Products of modules

This file defines constructors for linear maps whose domains or codomains are products.

It contains theorems relating these to each other, as well as to Submodule.prod, Submodule.map, Submodule.comap, LinearMap.range, and LinearMap.ker.

Main definitions

• products in the domain:
  • LinearMap.fst
  • LinearMap.snd
  • LinearMap.coprod
  • LinearMap.prod_ext
• products in the codomain:
  • LinearMap.inl
  • LinearMap.inr
  • LinearMap.prod
• products in both domain and codomain:
  • LinearMap.prodMap
  • LinearEquiv.prodMap
  • LinearEquiv.skewProd

def LinearMap.fst (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : M × M₂ →ₗ[R] M

The first projection of a product is a linear map.

Equations

• LinearMap.fst R M M₂ = { toFun := Prod.fst, map_add' := ⋯, map_smul' := ⋯ }

def LinearMap.snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : M × M₂ →ₗ[R] M₂

The second projection of a product is a linear map.

Equations

• LinearMap.snd R M M₂ = { toFun := Prod.snd, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem LinearMap.fst_apply {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : M × M₂) : (LinearMap.fst R M M₂) x = x.1

@[simp]

theorem LinearMap.snd_apply {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : M × M₂) : (LinearMap.snd R M M₂) x = x.2

theorem LinearMap.fst_surjective {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Function.Surjective ⇑(LinearMap.fst R M M₂)

theorem LinearMap.snd_surjective {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Function.Surjective ⇑(LinearMap.snd R M M₂)

def LinearMap.prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : M →ₗ[R] M₂ × M₃

The prod of two linear maps is a linear map.

Equations

• f.prod g = { toFun := Pi.prod ⇑f ⇑g, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem LinearMap.prod_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) (i : M) : (f.prod g) i = Pi.prod (⇑f) (⇑g) i

theorem LinearMap.coe_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : ⇑(f.prod g) = Pi.prod ⇑f ⇑g

@[simp]

theorem LinearMap.fst_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : LinearMap.fst R M₂ M₃ ∘ₗ f.prod g = f

@[simp]

theorem LinearMap.snd_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : LinearMap.snd R M₂ M₃ ∘ₗ f.prod g = g

@[simp]

theorem LinearMap.pair_fst_snd {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : (LinearMap.fst R M M₂).prod (LinearMap.snd R M M₂) = LinearMap.id

theorem LinearMap.prod_comp {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M₂ →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) (h : M →ₗ[R] M₂) : f.prod g ∘ₗ h = (f ∘ₗ h).prod (g ∘ₗ h)

def LinearMap.prodEquiv {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] [Module S M₂] [Module S M₃] [SMulCommClass R S M₂] [SMulCommClass R S M₃] : ((M →ₗ[R] M₂) × (M →ₗ[R] M₃)) ≃ₗ[S] M →ₗ[R] M₂ × M₃

Taking the product of two maps with the same domain is equivalent to taking the product of their codomains.

See note [bundled maps over different rings] for why separate R and S semirings are used.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LinearMap.prodEquiv_symm_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] [Module S M₂] [Module S M₃] [SMulCommClass R S M₂] [SMulCommClass R S M₃] (f : M →ₗ[R] M₂ × M₃) : (LinearMap.prodEquiv S).symm f = (LinearMap.fst R M₂ M₃ ∘ₗ f, LinearMap.snd R M₂ M₃ ∘ₗ f)

@[simp]

theorem LinearMap.prodEquiv_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] [Module S M₂] [Module S M₃] [SMulCommClass R S M₂] [SMulCommClass R S M₃] (f : (M →ₗ[R] M₂) × (M →ₗ[R] M₃)) : (LinearMap.prodEquiv S) f = f.1.prod f.2

def LinearMap.inl (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : M →ₗ[R] M × M₂

The left injection into a product is a linear map.

Equations

• LinearMap.inl R M M₂ = LinearMap.id.prod 0

def LinearMap.inr (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : M₂ →ₗ[R] M × M₂

The right injection into a product is a linear map.

Equations

• LinearMap.inr R M M₂ = LinearMap.prod 0 LinearMap.id

theorem LinearMap.range_inl (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.range (LinearMap.inl R M M₂) = LinearMap.ker (LinearMap.snd R M M₂)

theorem LinearMap.ker_snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.ker (LinearMap.snd R M M₂) = LinearMap.range (LinearMap.inl R M M₂)

theorem LinearMap.range_inr (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.range (LinearMap.inr R M M₂) = LinearMap.ker (LinearMap.fst R M M₂)

theorem LinearMap.ker_fst (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.ker (LinearMap.fst R M M₂) = LinearMap.range (LinearMap.inr R M M₂)

@[simp]

theorem LinearMap.fst_comp_inl (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.fst R M M₂ ∘ₗ LinearMap.inl R M M₂ = LinearMap.id

@[simp]

theorem LinearMap.snd_comp_inl (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.snd R M M₂ ∘ₗ LinearMap.inl R M M₂ = 0

@[simp]

theorem LinearMap.fst_comp_inr (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.fst R M M₂ ∘ₗ LinearMap.inr R M M₂ = 0

@[simp]

theorem LinearMap.snd_comp_inr (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.snd R M M₂ ∘ₗ LinearMap.inr R M M₂ = LinearMap.id

@[simp]

theorem LinearMap.coe_inl {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : ⇑(LinearMap.inl R M M₂) = fun (x : M) => (x, 0)

theorem LinearMap.inl_apply {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : M) : (LinearMap.inl R M M₂) x = (x, 0)

@[simp]

theorem LinearMap.coe_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : ⇑(LinearMap.inr R M M₂) = Prod.mk 0

theorem LinearMap.inr_apply {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : M₂) : (LinearMap.inr R M M₂) x = (0, x)

theorem LinearMap.inl_eq_prod {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.inl R M M₂ = LinearMap.id.prod 0

theorem LinearMap.inr_eq_prod {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.inr R M M₂ = LinearMap.prod 0 LinearMap.id

theorem LinearMap.inl_injective {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Function.Injective ⇑(LinearMap.inl R M M₂)

theorem LinearMap.inr_injective {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Function.Injective ⇑(LinearMap.inr R M M₂)

def LinearMap.coprod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : M × M₂ →ₗ[R] M₃

The coprod function x : M × M₂ ↦ f x.1 + g x.2 is a linear map.

Equations

• f.coprod g = f ∘ₗ LinearMap.fst R M M₂ + g ∘ₗ LinearMap.snd R M M₂

@[simp]

theorem LinearMap.coprod_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (x : M × M₂) : (f.coprod g) x = f x.1 + g x.2

@[simp]

theorem LinearMap.coprod_inl {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : f.coprod g ∘ₗ LinearMap.inl R M M₂ = f

@[simp]

theorem LinearMap.coprod_inr {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : f.coprod g ∘ₗ LinearMap.inr R M M₂ = g

@[simp]

theorem LinearMap.coprod_inl_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : (LinearMap.inl R M M₂).coprod (LinearMap.inr R M M₂) = LinearMap.id

theorem LinearMap.coprod_zero_left {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (g : M₂ →ₗ[R] M₃) : LinearMap.coprod 0 g = g ∘ₗ LinearMap.snd R M M₂

theorem LinearMap.coprod_zero_right {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) : f.coprod 0 = f ∘ₗ LinearMap.fst R M M₂

theorem LinearMap.comp_coprod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M₃ →ₗ[R] M₄) (g₁ : M →ₗ[R] M₃) (g₂ : M₂ →ₗ[R] M₃) : f ∘ₗ g₁.coprod g₂ = (f ∘ₗ g₁).coprod (f ∘ₗ g₂)

theorem LinearMap.fst_eq_coprod {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.fst R M M₂ = LinearMap.id.coprod 0

theorem LinearMap.snd_eq_coprod {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.snd R M M₂ = LinearMap.coprod 0 LinearMap.id

@[simp]

theorem LinearMap.coprod_comp_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M₂ →ₗ[R] M₄) (g : M₃ →ₗ[R] M₄) (f' : M →ₗ[R] M₂) (g' : M →ₗ[R] M₃) : f.coprod g ∘ₗ f'.prod g' = f ∘ₗ f' + g ∘ₗ g'

@[simp]

theorem LinearMap.coprod_map_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (S : Submodule R M) (S' : Submodule R M₂) : Submodule.map (f.coprod g) (S.prod S') = Submodule.map f S ⊔ Submodule.map g S'

def LinearMap.coprodEquiv {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] [Module S M₃] [SMulCommClass R S M₃] : ((M →ₗ[R] M₃) × (M₂ →ₗ[R] M₃)) ≃ₗ[S] M × M₂ →ₗ[R] M₃

Taking the product of two maps with the same codomain is equivalent to taking the product of their domains.

See note [bundled maps over different rings] for why separate R and S semirings are used.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LinearMap.coprodEquiv_symm_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] [Module S M₃] [SMulCommClass R S M₃] (f : M × M₂ →ₗ[R] M₃) : (LinearMap.coprodEquiv S).symm f = (f ∘ₗ LinearMap.inl R M M₂, f ∘ₗ LinearMap.inr R M M₂)

@[simp]

theorem LinearMap.coprodEquiv_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] [Module S M₃] [SMulCommClass R S M₃] (f : (M →ₗ[R] M₃) × (M₂ →ₗ[R] M₃)) : (LinearMap.coprodEquiv S) f = f.1.coprod f.2

theorem LinearMap.prod_ext_iff {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] {f g : M × M₂ →ₗ[R] M₃} : f = g ↔ f ∘ₗ LinearMap.inl R M M₂ = g ∘ₗ LinearMap.inl R M M₂ ∧ f ∘ₗ LinearMap.inr R M M₂ = g ∘ₗ LinearMap.inr R M M₂

theorem LinearMap.prod_ext {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] {f g : M × M₂ →ₗ[R] M₃} (hl : f ∘ₗ LinearMap.inl R M M₂ = g ∘ₗ LinearMap.inl R M M₂) (hr : f ∘ₗ LinearMap.inr R M M₂ = g ∘ₗ LinearMap.inr R M M₂) : f = g

Split equality of linear maps from a product into linear maps over each component, to allow ext to apply lemmas specific to M →ₗ M₃ and M₂ →ₗ M₃.

See note [partially-applied ext lemmas].

def LinearMap.prodMap {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) : M × M₂ →ₗ[R] M₃ × M₄

prod.map of two linear maps.

Equations

• f.prodMap g = (f ∘ₗ LinearMap.fst R M M₂).prod (g ∘ₗ LinearMap.snd R M M₂)

theorem LinearMap.coe_prodMap {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) : ⇑(f.prodMap g) = Prod.map ⇑f ⇑g

@[simp]

theorem LinearMap.prodMap_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) (x : M × M₂) : (f.prodMap g) x = (f x.1, g x.2)

theorem LinearMap.prodMap_comap_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄) (S : Submodule R M₂) (S' : Submodule R M₄) : Submodule.comap (f.prodMap g) (S.prod S') = (Submodule.comap f S).prod (Submodule.comap g S')

theorem LinearMap.ker_prodMap {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f : M →ₗ[R] M₂) (g : M₃ →ₗ[R] M₄) : LinearMap.ker (f.prodMap g) = (LinearMap.ker f).prod (LinearMap.ker g)

@[simp]

theorem LinearMap.prodMap_id {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.id.prodMap LinearMap.id = LinearMap.id

@[simp]

theorem LinearMap.prodMap_one {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.prodMap 1 1 = 1

theorem LinearMap.prodMap_comp {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} {M₅ : Type u_1} {M₆ : Type u_2} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [AddCommMonoid M₅] [AddCommMonoid M₆] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [Module R M₅] [Module R M₆] (f₁₂ : M →ₗ[R] M₂) (f₂₃ : M₂ →ₗ[R] M₃) (g₁₂ : M₄ →ₗ[R] M₅) (g₂₃ : M₅ →ₗ[R] M₆) : f₂₃.prodMap g₂₃ ∘ₗ f₁₂.prodMap g₁₂ = (f₂₃ ∘ₗ f₁₂).prodMap (g₂₃ ∘ₗ g₁₂)

theorem LinearMap.prodMap_mul {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f₁₂ f₂₃ : M →ₗ[R] M) (g₁₂ g₂₃ : M₂ →ₗ[R] M₂) : f₂₃.prodMap g₂₃ * f₁₂.prodMap g₁₂ = (f₂₃ * f₁₂).prodMap (g₂₃ * g₁₂)

theorem LinearMap.prodMap_add {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (f₁ f₂ : M →ₗ[R] M₃) (g₁ g₂ : M₂ →ₗ[R] M₄) : (f₁ + f₂).prodMap (g₁ + g₂) = f₁.prodMap g₁ + f₂.prodMap g₂

@[simp]

theorem LinearMap.prodMap_zero {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] : LinearMap.prodMap 0 0 = 0

@[simp]

theorem LinearMap.prodMap_smul {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [Module S M₃] [Module S M₄] [SMulCommClass R S M₃] [SMulCommClass R S M₄] (s : S) (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₄) : (s • f).prodMap (s • g) = s • f.prodMap g

def LinearMap.prodMapLinear (R : Type u) (M : Type v) (M₂ : Type w) (M₃ : Type y) (M₄ : Type z) (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [Module S M₃] [Module S M₄] [SMulCommClass R S M₃] [SMulCommClass R S M₄] : (M →ₗ[R] M₃) × (M₂ →ₗ[R] M₄) →ₗ[S] M × M₂ →ₗ[R] M₃ × M₄

LinearMap.prodMap as a LinearMap

Equations

• LinearMap.prodMapLinear R M M₂ M₃ M₄ S = { toFun := fun (f : (M →ₗ[R] M₃) × (M₂ →ₗ[R] M₄)) => f.1.prodMap f.2, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem LinearMap.prodMapLinear_apply (R : Type u) (M : Type v) (M₂ : Type w) (M₃ : Type y) (M₄ : Type z) (S : Type u_3) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] [Module S M₃] [Module S M₄] [SMulCommClass R S M₃] [SMulCommClass R S M₄] (f : (M →ₗ[R] M₃) × (M₂ →ₗ[R] M₄)) : (LinearMap.prodMapLinear R M M₂ M₃ M₄ S) f = f.1.prodMap f.2

def LinearMap.prodMapRingHom (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : (M →ₗ[R] M) × (M₂ →ₗ[R] M₂) →+* M × M₂ →ₗ[R] M × M₂

LinearMap.prodMap as a RingHom

Equations

• LinearMap.prodMapRingHom R M M₂ = { toFun := fun (f : (M →ₗ[R] M) × (M₂ →ₗ[R] M₂)) => f.1.prodMap f.2, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem LinearMap.prodMapRingHom_apply (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : (M →ₗ[R] M) × (M₂ →ₗ[R] M₂)) : (LinearMap.prodMapRingHom R M M₂) f = f.1.prodMap f.2

theorem LinearMap.inl_map_mul {R : Type u} [Semiring R] {A : Type u_4} [NonUnitalNonAssocSemiring A] [Module R A] {B : Type u_5} [NonUnitalNonAssocSemiring B] [Module R B] (a₁ a₂ : A) : (LinearMap.inl R A B) (a₁ * a₂) = (LinearMap.inl R A B) a₁ * (LinearMap.inl R A B) a₂

theorem LinearMap.inr_map_mul {R : Type u} [Semiring R] {A : Type u_4} [NonUnitalNonAssocSemiring A] [Module R A] {B : Type u_5} [NonUnitalNonAssocSemiring B] [Module R B] (b₁ b₂ : B) : (LinearMap.inr R A B) (b₁ * b₂) = (LinearMap.inr R A B) b₁ * (LinearMap.inr R A B) b₂

def LinearMap.prodMapAlgHom (R : Type u) (M : Type v) (M₂ : Type w) [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Module.End R M × Module.End R M₂ →ₐ[R] Module.End R (M × M₂)

LinearMap.prodMap as an AlgHom

Equations

• LinearMap.prodMapAlgHom R M M₂ = { toRingHom := LinearMap.prodMapRingHom R M M₂, commutes' := ⋯ }

@[simp]

theorem LinearMap.prodMapAlgHom_apply_apply (R : Type u) (M : Type v) (M₂ : Type w) [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : (M →ₗ[R] M) × (M₂ →ₗ[R] M₂)) (i : M × M₂) : ((LinearMap.prodMapAlgHom R M M₂) f) i = (f.1 i.1, f.2 i.2)

theorem LinearMap.range_coprod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : LinearMap.range (f.coprod g) = LinearMap.range f ⊔ LinearMap.range g

theorem LinearMap.isCompl_range_inl_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : IsCompl (LinearMap.range (LinearMap.inl R M M₂)) (LinearMap.range (LinearMap.inr R M M₂))

theorem LinearMap.sup_range_inl_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.range (LinearMap.inl R M M₂) ⊔ LinearMap.range (LinearMap.inr R M M₂) = ⊤

theorem LinearMap.disjoint_inl_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Disjoint (LinearMap.range (LinearMap.inl R M M₂)) (LinearMap.range (LinearMap.inr R M M₂))

theorem LinearMap.map_coprod_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (p : Submodule R M) (q : Submodule R M₂) : Submodule.map (f.coprod g) (p.prod q) = Submodule.map f p ⊔ Submodule.map g q

theorem LinearMap.comap_prod_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) (p : Submodule R M₂) (q : Submodule R M₃) : Submodule.comap (f.prod g) (p.prod q) = Submodule.comap f p ⊓ Submodule.comap g q

theorem LinearMap.prod_eq_inf_comap {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) (q : Submodule R M₂) : p.prod q = Submodule.comap (LinearMap.fst R M M₂) p ⊓ Submodule.comap (LinearMap.snd R M M₂) q

theorem LinearMap.prod_eq_sup_map {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) (q : Submodule R M₂) : p.prod q = Submodule.map (LinearMap.inl R M M₂) p ⊔ Submodule.map (LinearMap.inr R M M₂) q

theorem LinearMap.span_inl_union_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {s : Set M} {t : Set M₂} : Submodule.span R (⇑(LinearMap.inl R M M₂) '' s ∪ ⇑(LinearMap.inr R M M₂) '' t) = (Submodule.span R s).prod (Submodule.span R t)

@[simp]

theorem LinearMap.ker_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : LinearMap.ker (f.prod g) = LinearMap.ker f ⊓ LinearMap.ker g

theorem LinearMap.range_prod_le {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R M₂] [Module R M₃] (f : M →ₗ[R] M₂) (g : M →ₗ[R] M₃) : LinearMap.range (f.prod g) ≤ (LinearMap.range f).prod (LinearMap.range g)

theorem LinearMap.ker_prod_ker_le_ker_coprod {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {M₂ : Type u_3} [AddCommGroup M₂] [Module R M₂] {M₃ : Type u_4} [AddCommGroup M₃] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : (LinearMap.ker f).prod (LinearMap.ker g) ≤ LinearMap.ker (f.coprod g)

theorem LinearMap.ker_coprod_of_disjoint_range {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {M₂ : Type u_3} [AddCommGroup M₂] [Module R M₂] {M₃ : Type u_4} [AddCommGroup M₃] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (hd : Disjoint (LinearMap.range f) (LinearMap.range g)) : LinearMap.ker (f.coprod g) = (LinearMap.ker f).prod (LinearMap.ker g)

theorem Submodule.sup_eq_range {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] (p q : Submodule R M) : p ⊔ q = LinearMap.range (p.subtype.coprod q.subtype)

@[simp]

theorem Submodule.map_inl {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) : Submodule.map (LinearMap.inl R M M₂) p = p.prod ⊥

@[simp]

theorem Submodule.map_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (q : Submodule R M₂) : Submodule.map (LinearMap.inr R M M₂) q = ⊥.prod q

@[simp]

theorem Submodule.comap_fst {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) : Submodule.comap (LinearMap.fst R M M₂) p = p.prod ⊤

@[simp]

theorem Submodule.comap_snd {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (q : Submodule R M₂) : Submodule.comap (LinearMap.snd R M M₂) q = ⊤.prod q

@[simp]

theorem Submodule.prod_comap_inl {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) (q : Submodule R M₂) : Submodule.comap (LinearMap.inl R M M₂) (p.prod q) = p

@[simp]

theorem Submodule.prod_comap_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) (q : Submodule R M₂) : Submodule.comap (LinearMap.inr R M M₂) (p.prod q) = q

@[simp]

theorem Submodule.prod_map_fst {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) (q : Submodule R M₂) : Submodule.map (LinearMap.fst R M M₂) (p.prod q) = p

@[simp]

theorem Submodule.prod_map_snd {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (p : Submodule R M) (q : Submodule R M₂) : Submodule.map (LinearMap.snd R M M₂) (p.prod q) = q

@[simp]

theorem Submodule.ker_inl {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.ker (LinearMap.inl R M M₂) = ⊥

@[simp]

theorem Submodule.ker_inr {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.ker (LinearMap.inr R M M₂) = ⊥

@[simp]

theorem Submodule.range_fst {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.range (LinearMap.fst R M M₂) = ⊤

@[simp]

theorem Submodule.range_snd {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.range (LinearMap.snd R M M₂) = ⊤

def Submodule.fst (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule R (M × M₂)

M as a submodule of M × N.

Equations

• Submodule.fst R M M₂ = Submodule.comap (LinearMap.snd R M M₂) ⊥

def Submodule.fstEquiv (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : ↥(Submodule.fst R M M₂) ≃ₗ[R] M

M as a submodule of M × N is isomorphic to M.

Equations

• Submodule.fstEquiv R M M₂ = { toFun := fun (x : ↥(Submodule.fst R M M₂)) => (↑x).1, map_add' := ⋯, map_smul' := ⋯, invFun := fun (m : M) => ⟨(m, 0), ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Submodule.fstEquiv_symm_apply_coe (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (m : M) : ↑((Submodule.fstEquiv R M M₂).symm m) = (m, 0)

@[simp]

theorem Submodule.fstEquiv_apply (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : ↥(Submodule.fst R M M₂)) : (Submodule.fstEquiv R M M₂) x = (↑x).1

theorem Submodule.fst_map_fst (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule.map (LinearMap.fst R M M₂) (Submodule.fst R M M₂) = ⊤

theorem Submodule.fst_map_snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule.map (LinearMap.snd R M M₂) (Submodule.fst R M M₂) = ⊥

def Submodule.snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule R (M × M₂)

N as a submodule of M × N.

Equations

• Submodule.snd R M M₂ = Submodule.comap (LinearMap.fst R M M₂) ⊥

def Submodule.sndEquiv (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : ↥(Submodule.snd R M M₂) ≃ₗ[R] M₂

N as a submodule of M × N is isomorphic to N.

Equations

• Submodule.sndEquiv R M M₂ = { toFun := fun (x : ↥(Submodule.snd R M M₂)) => (↑x).2, map_add' := ⋯, map_smul' := ⋯, invFun := fun (n : M₂) => ⟨(0, n), ⋯⟩, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem Submodule.sndEquiv_apply (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : ↥(Submodule.snd R M M₂)) : (Submodule.sndEquiv R M M₂) x = (↑x).2

@[simp]

theorem Submodule.sndEquiv_symm_apply_coe (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (n : M₂) : ↑((Submodule.sndEquiv R M M₂).symm n) = (0, n)

theorem Submodule.snd_map_fst (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule.map (LinearMap.fst R M M₂) (Submodule.snd R M M₂) = ⊥

theorem Submodule.snd_map_snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule.map (LinearMap.snd R M M₂) (Submodule.snd R M M₂) = ⊤

theorem Submodule.fst_sup_snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule.fst R M M₂ ⊔ Submodule.snd R M M₂ = ⊤

theorem Submodule.fst_inf_snd (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : Submodule.fst R M M₂ ⊓ Submodule.snd R M M₂ = ⊥

theorem Submodule.le_prod_iff (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {p₁ : Submodule R M} {p₂ : Submodule R M₂} {q : Submodule R (M × M₂)} : q ≤ p₁.prod p₂ ↔ Submodule.map (LinearMap.fst R M M₂) q ≤ p₁ ∧ Submodule.map (LinearMap.snd R M M₂) q ≤ p₂

theorem Submodule.prod_le_iff (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {p₁ : Submodule R M} {p₂ : Submodule R M₂} {q : Submodule R (M × M₂)} : p₁.prod p₂ ≤ q ↔ Submodule.map (LinearMap.inl R M M₂) p₁ ≤ q ∧ Submodule.map (LinearMap.inr R M M₂) p₂ ≤ q

theorem Submodule.prod_eq_bot_iff (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {p₁ : Submodule R M} {p₂ : Submodule R M₂} : p₁.prod p₂ = ⊥ ↔ p₁ = ⊥ ∧ p₂ = ⊥

theorem Submodule.prod_eq_top_iff (R : Type u) (M : Type v) (M₂ : Type w) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] {p₁ : Submodule R M} {p₂ : Submodule R M₂} : p₁.prod p₂ = ⊤ ↔ p₁ = ⊤ ∧ p₂ = ⊤

def LinearEquiv.prodComm (R : Type u_3) (M : Type u_4) (N : Type u_5) [Semiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] : (M × N) ≃ₗ[R] N × M

Product of modules is commutative up to linear isomorphism.

Equations

• LinearEquiv.prodComm R M N = { toFun := Prod.swap, map_add' := ⋯, map_smul' := ⋯, invFun := AddEquiv.prodComm.invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LinearEquiv.prodComm_apply (R : Type u_3) (M : Type u_4) (N : Type u_5) [Semiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (a✝ : M × N) : (LinearEquiv.prodComm R M N) a✝ = a✝.swap

theorem LinearEquiv.fst_comp_prodComm {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.fst R M₂ M ∘ₗ ↑(LinearEquiv.prodComm R M M₂) = LinearMap.snd R M M₂

theorem LinearEquiv.snd_comp_prodComm {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] : LinearMap.snd R M₂ M ∘ₗ ↑(LinearEquiv.prodComm R M M₂) = LinearMap.fst R M M₂

def LinearEquiv.prodAssoc (R : Type u_3) (M₁ : Type u_4) (M₂ : Type u_5) (M₃ : Type u_6) [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₁] [Module R M₂] [Module R M₃] : ((M₁ × M₂) × M₃) ≃ₗ[R] M₁ × M₂ × M₃

Product of modules is associative up to linear isomorphism.

Equations

• LinearEquiv.prodAssoc R M₁ M₂ M₃ = { toFun := AddEquiv.prodAssoc.toFun, map_add' := ⋯, map_smul' := ⋯, invFun := AddEquiv.prodAssoc.invFun, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem LinearEquiv.prodAssoc_apply (R : Type u_3) (M₁ : Type u_4) (M₂ : Type u_5) (M₃ : Type u_6) [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₁] [Module R M₂] [Module R M₃] (a✝ : (M₁ × M₂) × M₃) : (LinearEquiv.prodAssoc R M₁ M₂ M₃) a✝ = AddEquiv.prodAssoc.toFun a✝

theorem LinearEquiv.fst_comp_prodAssoc {R : Type u} {M₂ : Type w} {M₃ : Type y} {M₁ : Type u_3} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₁] [Module R M₂] [Module R M₃] : LinearMap.fst R M₁ (M₂ × M₃) ∘ₗ ↑(LinearEquiv.prodAssoc R M₁ M₂ M₃) = LinearMap.fst R M₁ M₂ ∘ₗ LinearMap.fst R (M₁ × M₂) M₃

theorem LinearEquiv.snd_comp_prodAssoc {R : Type u} {M₂ : Type w} {M₃ : Type y} {M₁ : Type u_3} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₁] [Module R M₂] [Module R M₃] : LinearMap.snd R M₁ (M₂ × M₃) ∘ₗ ↑(LinearEquiv.prodAssoc R M₁ M₂ M₃) = (LinearMap.snd R M₁ M₂).prodMap LinearMap.id

def LinearEquiv.prodProdProdComm (R : Type u) (M : Type v) (M₂ : Type w) (M₃ : Type y) (M₄ : Type z) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] : ((M × M₂) × M₃ × M₄) ≃ₗ[R] (M × M₃) × M₂ × M₄

Four-way commutativity of prod. The name matches mul_mul_mul_comm.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LinearEquiv.prodProdProdComm_apply (R : Type u) (M : Type v) (M₂ : Type w) (M₃ : Type y) (M₄ : Type z) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] (mnmn : (M × M₂) × M₃ × M₄) : (LinearEquiv.prodProdProdComm R M M₂ M₃ M₄) mnmn = ((mnmn.1.1, mnmn.2.1), mnmn.1.2, mnmn.2.2)

@[simp]

theorem LinearEquiv.prodProdProdComm_symm (R : Type u) (M : Type v) (M₂ : Type w) (M₃ : Type y) (M₄ : Type z) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] : (LinearEquiv.prodProdProdComm R M M₂ M₃ M₄).symm = LinearEquiv.prodProdProdComm R M M₃ M₂ M₄

@[simp]

theorem LinearEquiv.prodProdProdComm_toAddEquiv (R : Type u) (M : Type v) (M₂ : Type w) (M₃ : Type y) (M₄ : Type z) [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] [Module R M] [Module R M₂] [Module R M₃] [Module R M₄] : ↑(LinearEquiv.prodProdProdComm R M M₂ M₃ M₄) = AddEquiv.prodProdProdComm M M₂ M₃ M₄

def LinearEquiv.prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) : (M × M₃) ≃ₗ[R] M₂ × M₄

Product of linear equivalences; the maps come from Equiv.prodCongr.

Equations

• e₁.prod e₂ = { toFun := (e₁.toAddEquiv.prodCongr e₂.toAddEquiv).toFun, map_add' := ⋯, map_smul' := ⋯, invFun := (e₁.toAddEquiv.prodCongr e₂.toAddEquiv).invFun, left_inv := ⋯, right_inv := ⋯ }

theorem LinearEquiv.prod_symm {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) : (e₁.prod e₂).symm = e₁.symm.prod e₂.symm

@[simp]

theorem LinearEquiv.prod_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) (p : M × M₃) : (e₁.prod e₂) p = (e₁ p.1, e₂ p.2)

@[simp]

theorem LinearEquiv.coe_prod {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) : ↑(e₁.prod e₂) = (↑e₁).prodMap ↑e₂

def LinearEquiv.skewProd {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommGroup M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) (f : M →ₗ[R] M₄) : (M × M₃) ≃ₗ[R] M₂ × M₄

Equivalence given by a block lower diagonal matrix. e₁ and e₂ are diagonal square blocks, and f is a rectangular block below the diagonal.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem LinearEquiv.skewProd_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommGroup M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) (f : M →ₗ[R] M₄) (x : M × M₃) : (e₁.skewProd e₂ f) x = (e₁ x.1, e₂ x.2 + f x.1)

@[simp]

theorem LinearEquiv.skewProd_symm_apply {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommGroup M₄] {module_M : Module R M} {module_M₂ : Module R M₂} {module_M₃ : Module R M₃} {module_M₄ : Module R M₄} (e₁ : M ≃ₗ[R] M₂) (e₂ : M₃ ≃ₗ[R] M₄) (f : M →ₗ[R] M₄) (x : M₂ × M₄) : (e₁.skewProd e₂ f).symm x = (e₁.symm x.1, e₂.symm (x.2 - f (e₁.symm x.1)))

theorem LinearMap.range_prod_eq {R : Type u} {M : Type v} {M₂ : Type w} {M₃ : Type y} [Ring R] [AddCommGroup M] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M] [Module R M₂] [Module R M₃] {f : M →ₗ[R] M₂} {g : M →ₗ[R] M₃} (h : LinearMap.ker f ⊔ LinearMap.ker g = ⊤) : LinearMap.range (f.prod g) = (LinearMap.range f).prod (LinearMap.range g)

If the union of the kernels ker f and ker g spans the domain, then the range of Prod f g is equal to the product of range f and range g.

Tunnels and tailings

NOTE: The proof of strong rank condition for noetherian rings is changed. LinearMap.tunnel and LinearMap.tailing are not used in mathlib anymore. These are marked as deprecated with no replacements. If you use them in external projects, please consider using other arguments instead.

Some preliminary work for establishing the strong rank condition for noetherian rings.

Given a morphism f : M × N →ₗ[R] M which is i : Injective f, we can find an infinite decreasing tunnel f i n of copies of M inside M, and sitting beside these, an infinite sequence of copies of N.

We picturesquely name these as tailing f i n for each individual copy of N, and tailings f i n for the supremum of the first n+1 copies: they are the pieces left behind, sitting inside the tunnel.

By construction, each tailing f i (n+1) is disjoint from tailings f i n; later, when we assume M is noetherian, this implies that N must be trivial, and establishes the strong rank condition for any left-noetherian ring.

@[deprecated]

def LinearMap.tunnelAux {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (Kφ : (K : Submodule R M) × ↥K ≃ₗ[R] M) : M × N →ₗ[R] M

An auxiliary construction for tunnel. The composition of f, followed by the isomorphism back to K, followed by the inclusion of this submodule back into M.

Equations

• f.tunnelAux Kφ = (Kφ.fst.subtype ∘ₗ ↑Kφ.snd.symm) ∘ₗ f

@[deprecated]

theorem LinearMap.tunnelAux_injective {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (Kφ : (K : Submodule R M) × ↥K ≃ₗ[R] M) : Function.Injective ⇑(f.tunnelAux Kφ)

@[deprecated]

def LinearMap.tunnel' {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) : ℕ → (K : Submodule R M) × ↥K ≃ₗ[R] M

Auxiliary definition for tunnel.

Equations

• One or more equations did not get rendered due to their size.
• f.tunnel' i 0 = ⟨⊤, LinearEquiv.ofTop ⊤ ⋯⟩

@[deprecated]

def LinearMap.tunnel {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) : ℕ →o (Submodule R M)ᵒᵈ

Give an injective map f : M × N →ₗ[R] M we can find a nested sequence of submodules all isomorphic to M.

Equations

• f.tunnel i = { toFun := fun (n : ℕ) => OrderDual.toDual (f.tunnel' i n).fst, monotone' := ⋯ }

@[deprecated]

def LinearMap.tailing {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : Submodule R M

Give an injective map f : M × N →ₗ[R] M we can find a sequence of submodules all isomorphic to N.

Equations

• f.tailing i n = Submodule.map (f.tunnelAux (f.tunnel' i n)) (Submodule.snd R M N)

@[deprecated]

def LinearMap.tailingLinearEquiv {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : ↥(f.tailing i n) ≃ₗ[R] N

Each tailing f i n is a copy of N.

Equations

• f.tailingLinearEquiv i n = (Submodule.equivMapOfInjective (f.tunnelAux (f.tunnel' i n)) ⋯ (Submodule.snd R M N)).symm.trans (Submodule.sndEquiv R M N)

@[deprecated]

theorem LinearMap.tailing_le_tunnel {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : f.tailing i n ≤ OrderDual.ofDual ((f.tunnel i) n)

@[deprecated]

theorem LinearMap.tailing_disjoint_tunnel_succ {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : Disjoint (f.tailing i n) (OrderDual.ofDual ((f.tunnel i) (n + 1)))

@[deprecated]

theorem LinearMap.tailing_sup_tunnel_succ_le_tunnel {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : f.tailing i n ⊔ OrderDual.ofDual ((f.tunnel i) (n + 1)) ≤ OrderDual.ofDual ((f.tunnel i) n)

@[deprecated]

def LinearMap.tailings {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) : ℕ → Submodule R M

The supremum of all the copies of N found inside the tunnel.

Equations

• f.tailings i = ⇑(partialSups (f.tailing i))

@[simp, deprecated]

theorem LinearMap.tailings_zero {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) : f.tailings i 0 = f.tailing i 0

@[simp, deprecated]

theorem LinearMap.tailings_succ {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : f.tailings i (n + 1) = f.tailings i n ⊔ f.tailing i (n + 1)

@[deprecated]

theorem LinearMap.tailings_disjoint_tunnel {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : Disjoint (f.tailings i n) (OrderDual.ofDual ((f.tunnel i) (n + 1)))

@[deprecated]

theorem LinearMap.tailings_disjoint_tailing {R : Type u} {M : Type v} [Ring R] {N : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M × N →ₗ[R] M) (i : Function.Injective ⇑f) (n : ℕ) : Disjoint (f.tailings i n) (f.tailing i (n + 1))

def LinearMap.graph {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : M →ₗ[R] M₂) : Submodule R (M × M₂)

Graph of a linear map.

Equations

• f.graph = { carrier := {p : M × M₂ | p.2 = f p.1}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

@[simp]

theorem LinearMap.mem_graph_iff {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : M →ₗ[R] M₂) (x : M × M₂) : x ∈ f.graph ↔ x.2 = f x.1

theorem LinearMap.graph_eq_ker_coprod {R : Type u} {M₃ : Type y} {M₄ : Type z} [Semiring R] [AddCommGroup M₃] [AddCommGroup M₄] [Module R M₃] [Module R M₄] (g : M₃ →ₗ[R] M₄) : g.graph = LinearMap.ker ((-g).coprod LinearMap.id)

theorem LinearMap.graph_eq_range_prod {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (f : M →ₗ[R] M₂) : f.graph = LinearMap.range (LinearMap.id.prod f)

theorem LinearMap.exists_range_eq_graph {R : Type u_3} {S : Type u_4} {G : Type u_5} {H : Type u_6} {I : Type u_7} [Semiring R] [Semiring S] {σ : R →+* S} [RingHomSurjective σ] [AddCommMonoid G] [Module R G] [AddCommMonoid H] [Module S H] [AddCommMonoid I] [Module S I] {f : G →ₛₗ[σ] H × I} (hf₁ : Function.Surjective (Prod.fst ∘ ⇑f)) (hf : ∀ (g₁ g₂ : G), (f g₁).1 = (f g₂).1 → (f g₁).2 = (f g₂).2) : ∃ (f' : H →ₗ[S] I), LinearMap.range f = f'.graph

Vertical line test for module homomorphisms.

Let f : G → H × I be a linear (or semilinear) map to a product. Assume that f is surjective on the first factor and that the image of f intersects every "vertical line" {(h, i) | i : I} at most once. Then the image of f is the graph of some linear map f' : H → I.

theorem Submodule.exists_eq_graph {S : Type u_4} {H : Type u_6} {I : Type u_7} [Semiring S] [AddCommMonoid H] [Module S H] [AddCommMonoid I] [Module S I] {G : Submodule S (H × I)} (hf₁ : Function.Bijective (Prod.fst ∘ ⇑G.subtype)) : ∃ (f : H →ₗ[S] I), G = f.graph

Vertical line test for module homomorphisms.

Let G ≤ H × I be a submodule of a product of modules. Assume that G maps bijectively to the first factor. Then G is the graph of some module homomorphism f : H →ₗ[R] I.

theorem LinearMap.exists_linearEquiv_eq_graph {R : Type u_3} {S : Type u_4} {G : Type u_5} {H : Type u_6} {I : Type u_7} [Semiring R] [Semiring S] {σ : R →+* S} [RingHomSurjective σ] [AddCommMonoid G] [Module R G] [AddCommMonoid H] [Module S H] [AddCommMonoid I] [Module S I] {f : G →ₛₗ[σ] H × I} (hf₁ : Function.Surjective (Prod.fst ∘ ⇑f)) (hf₂ : Function.Surjective (Prod.snd ∘ ⇑f)) (hf : ∀ (g₁ g₂ : G), (f g₁).1 = (f g₂).1 ↔ (f g₁).2 = (f g₂).2) : ∃ (e : H ≃ₗ[S] I), LinearMap.range f = (↑e).graph

Line test for module isomorphisms.

Let f : G → H × I be a homomorphism to a product of modules. Assume that f is surjective onto both factors and that the image of f intersects every "vertical line" {(h, i) | i : I} and every "horizontal line" {(h, i) | h : H} at most once. Then the image of f is the graph of some module isomorphism f' : H ≃ I.

theorem Submodule.exists_equiv_eq_graph {S : Type u_4} {H : Type u_6} {I : Type u_7} [Semiring S] [AddCommMonoid H] [Module S H] [AddCommMonoid I] [Module S I] {G : Submodule S (H × I)} (hG₁ : Function.Bijective (Prod.fst ∘ ⇑G.subtype)) (hG₂ : Function.Bijective (Prod.snd ∘ ⇑G.subtype)) : ∃ (e : H ≃ₗ[S] I), G = (↑e).graph

Goursat's lemma for module isomorphisms.

Let G ≤ H × I be a submodule of a product of modules. Assume that the natural maps from G to both factors are bijective. Then G is the graph of some module isomorphism f : H ≃ I.