Extra facts about `Prod` #

This file proves various simple lemmas about Prod. It also defines better delaborators for product projections.

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theorem Prod.swap_eq_iff_eq_swap {α : Type u_1} {β : Type u_2} {x : α × β} {y : β × α} :

x.swap = y ↔ x = y.swap

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def Prod.mk.injArrow {α : Type u_1} {β : Type u_2} {x₁ : α} {y₁ : β} {x₂ : α} {y₂ : β} :

(x₁, y₁) = (x₂, y₂) → ⦃P : Sort u_5⦄ → (x₁ = x₂ → y₁ = y₂ → P) → P

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

Instances For

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@[simp]

theorem Prod.mk.eta {α : Type u_1} {β : Type u_2} {p : α × β} :

(p.1, p.2) = p

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theorem Prod.forall' {α : Type u_1} {β : Type u_2} {p : α → β → Prop} :

(∀ (x : α × β), p x.1 x.2) ↔ ∀ (a : α) (b : β), p a b

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theorem Prod.exists' {α : Type u_1} {β : Type u_2} {p : α → β → Prop} :

(∃ (x : α × β), p x.1 x.2) ↔ ∃ (a : α), ∃ (b : β), p a b

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@[simp]

theorem Prod.snd_comp_mk {α : Type u_1} {β : Type u_2} (x : α) :

snd ∘ mk x = id

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@[simp]

theorem Prod.fst_comp_mk {α : Type u_1} {β : Type u_2} (x : α) :

fst ∘ mk x = Function.const β x

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theorem Prod.map_apply' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ) (p : α × β) :

map f g p = (f p.1, g p.2)

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theorem Prod.map_fst' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ) :

fst ∘ map f g = f ∘ fst

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theorem Prod.map_snd' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ) :

snd ∘ map f g = g ∘ snd

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theorem Prod.mk_inj {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} :

(a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂

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theorem Prod.mk_right_injective {α : Type u_5} {β : Type u_6} (a : α) :

Function.Injective (mk a)

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theorem Prod.mk_left_injective {α : Type u_5} {β : Type u_6} (b : β) :

Function.Injective fun (a : α) => (a, b)

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theorem Prod.mk_right_inj {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} :

(a, b₁) = (a, b₂) ↔ b₁ = b₂

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theorem Prod.mk_left_inj {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} :

(a₁, b) = (a₂, b) ↔ a₁ = a₂

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theorem Prod.map_def {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ} {g : β → δ} :

map f g = fun (p : α × β) => (f p.1, g p.2)

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theorem Prod.id_prod {α : Type u_1} {β : Type u_2} :

(fun (p : α × β) => (p.1, p.2)) = id

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@[simp]

theorem Prod.map_iterate {α : Type u_1} {β : Type u_2} (f : α → α) (g : β → β) (n : ℕ) :

(map f g)^[n] = map f^[n] g^[n]

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theorem Prod.fst_surjective {α : Type u_1} {β : Type u_2} [h : Nonempty β] :

Function.Surjective fst

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theorem Prod.snd_surjective {α : Type u_1} {β : Type u_2} [h : Nonempty α] :

Function.Surjective snd

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theorem Prod.fst_injective {α : Type u_1} {β : Type u_2} [Subsingleton β] :

Function.Injective fst

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theorem Prod.snd_injective {α : Type u_1} {β : Type u_2} [Subsingleton α] :

Function.Injective snd

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@[simp]

theorem Prod.swap_leftInverse {α : Type u_1} {β : Type u_2} :

Function.LeftInverse swap swap

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@[simp]

theorem Prod.swap_rightInverse {α : Type u_1} {β : Type u_2} :

Function.RightInverse swap swap

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theorem Prod.swap_injective {α : Type u_1} {β : Type u_2} :

Function.Injective swap

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theorem Prod.swap_surjective {α : Type u_1} {β : Type u_2} :

Function.Surjective swap

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theorem Prod.swap_bijective {α : Type u_1} {β : Type u_2} :

Function.Bijective swap

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theorem Function.Semiconj.swap_map {α : Type u_1} {β : Type u_2} (f : α → α) (g : β → β) :

Semiconj Prod.swap (Prod.map f g) (Prod.map g f)

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theorem Prod.eq_iff_fst_eq_snd_eq {α : Type u_1} {β : Type u_2} {p q : α × β} :

p = q ↔ p.1 = q.1 ∧ p.2 = q.2

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theorem Prod.fst_eq_iff {α : Type u_1} {β : Type u_2} {p : α × β} {x : α} :

p.1 = x ↔ p = (x, p.2)

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theorem Prod.snd_eq_iff {α : Type u_1} {β : Type u_2} {p : α × β} {x : β} :

p.2 = x ↔ p = (p.1, x)

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theorem Prod.lex_iff {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {x y : α × β} :

Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2

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@[implicit_reducible]

instance Prod.Lex.decidable {α : Type u_1} {β : Type u_2} [DecidableEq α] (r : α → α → Prop) (s : β → β → Prop) [DecidableRel r] [DecidableRel s] :

DecidableRel (Prod.Lex r s)

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Prod.Lex.decidable r s x✝¹ x✝ = decidable_of_decidable_of_iff ⋯

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theorem Prod.Lex.refl_left {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Refl r] (x : α × β) :

Prod.Lex r s x x

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instance Prod.instReflLex_mathlib {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Refl r] :

Std.Refl (Prod.Lex r s)

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theorem Prod.Lex.refl_right {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Refl s] (x : α × β) :

Prod.Lex r s x x

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instance Prod.instReflLex_mathlib_1 {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Refl s] :

Std.Refl (Prod.Lex r s)

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instance Prod.instIrreflLex_mathlib {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Irrefl r] [Std.Irrefl s] :

Std.Irrefl (Prod.Lex r s)

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theorem Prod.Lex.trans {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrans α r] [IsTrans β s] {x y z : α × β} :

Prod.Lex r s x y → Prod.Lex r s y z → Prod.Lex r s x z

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instance Prod.instIsTransLex {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrans α r] [IsTrans β s] :

IsTrans (α × β) (Prod.Lex r s)

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instance Prod.instAntisymmLexOfIsStrictOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsStrictOrder α r] [Std.Antisymm s] :

Std.Antisymm (Prod.Lex r s)

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instance Prod.total_left {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Total r] :

Std.Total (Prod.Lex r s)

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instance Prod.total_right {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Trichotomous r] [Std.Total s] :

Std.Total (Prod.Lex r s)

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instance Prod.trichotomous {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Trichotomous r] [Std.Trichotomous s] :

Std.Trichotomous (Prod.Lex r s)

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instance Prod.instAsymmLex_mathlib {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [Std.Asymm r] [Std.Asymm s] :

Std.Asymm (Prod.Lex r s)

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theorem Function.Injective.prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ} {g : β → δ} (hf : Injective f) (hg : Injective g) :

Injective (Prod.map f g)

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theorem Function.Surjective.prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ} {g : β → δ} (hf : Surjective f) (hg : Surjective g) :

Surjective (Prod.map f g)

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theorem Function.Bijective.prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ} {g : β → δ} (hf : Bijective f) (hg : Bijective g) :

Bijective (Prod.map f g)

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theorem Function.LeftInverse.prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f₁ : α → β} {g₁ : γ → δ} {f₂ : β → α} {g₂ : δ → γ} (hf : LeftInverse f₁ f₂) (hg : LeftInverse g₁ g₂) :

LeftInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂)

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theorem Function.RightInverse.prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f₁ : α → β} {g₁ : γ → δ} {f₂ : β → α} {g₂ : δ → γ} :

RightInverse f₁ f₂ → RightInverse g₁ g₂ → RightInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂)

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theorem Function.Involutive.prodMap {α : Type u_1} {β : Type u_2} {f : α → α} {g : β → β} :

Involutive f → Involutive g → Involutive (Prod.map f g)

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@[simp]

theorem Prod.map_injective {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [Nonempty α] [Nonempty β] {f : α → γ} {g : β → δ} :

Function.Injective (map f g) ↔ Function.Injective f ∧ Function.Injective g

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@[simp]

theorem Prod.map_surjective {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [Nonempty γ] [Nonempty δ] {f : α → γ} {g : β → δ} :

Function.Surjective (map f g) ↔ Function.Surjective f ∧ Function.Surjective g

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@[simp]

theorem Prod.map_bijective {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [Nonempty α] [Nonempty β] {f : α → γ} {g : β → δ} :

Function.Bijective (map f g) ↔ Function.Bijective f ∧ Function.Bijective g

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@[simp]

theorem Prod.map_leftInverse {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [Nonempty β] [Nonempty δ] {f₁ : α → β} {g₁ : γ → δ} {f₂ : β → α} {g₂ : δ → γ} :

Function.LeftInverse (map f₁ g₁) (map f₂ g₂) ↔ Function.LeftInverse f₁ f₂ ∧ Function.LeftInverse g₁ g₂

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@[simp]

theorem Prod.map_rightInverse {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [Nonempty α] [Nonempty γ] {f₁ : α → β} {g₁ : γ → δ} {f₂ : β → α} {g₂ : δ → γ} :