Extra facts about `Prod` #

This file defines Prod.swap : α × β → β × α and proves various simple lemmas about Prod. It also defines better delaborators for product projections.

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@[deprecated Prod.map_apply]

theorem Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → β) (g : γ → δ) (x : α) (y : γ) :

Prod.map f g (x, y) = (f x, g y)

Alias of Prod.map_apply.

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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

Equations

Prod.mk.injArrow h₁ h₂ = Prod.noConfusion h₁ h₂

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) :

Prod.snd ∘ Prod.mk x = id

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@[simp]

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

Prod.fst ∘ Prod.mk x = Function.const β x

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@[simp]

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

Prod.map f g (a, b) = (f a, g b)

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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) :

Prod.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) :

Prod.fst ∘ Prod.map f g = f ∘ Prod.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) :

Prod.snd ∘ Prod.map f g = g ∘ Prod.snd

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theorem Prod.map_comp_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} {ζ : Type u_6} (f : α → β) (f' : γ → δ) (g : β → ε) (g' : δ → ζ) :

Prod.map g g' ∘ Prod.map f f' = Prod.map (g ∘ f) (g' ∘ f')

Composing a Prod.map with another Prod.map is equal to a single Prod.map of composed functions.

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theorem Prod.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5} {ζ : Type u_6} (f : α → β) (f' : γ → δ) (g : β → ε) (g' : δ → ζ) (x : α × γ) :

Prod.map g g' (Prod.map f f' x) = Prod.map (g ∘ f) (g' ∘ f') x

Composing a Prod.map with another Prod.map is equal to a single Prod.map of composed functions, fully applied.

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

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

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

Function.Injective (Prod.mk a)

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

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

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

(a, b₁) = (a, b₂) ↔ b₁ = b₂

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theorem Prod.mk_inj_right {α : Type u_1} {β : Type u_2} {a₁ : α} {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} :

Prod.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_id {α : Type u_1} {β : Type u_2} :

Prod.map id id = id

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@[simp]

theorem Prod.map_id' {α : Type u_1} {β : Type u_2} :

(Prod.map (fun (a : α) => a) fun (b : β) => b) = fun (x : α × β) => x

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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) :

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

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@[deprecated Prod.map_iterate]

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

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

Alias of Prod.map_iterate.

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

Function.Surjective Prod.fst

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

Function.Surjective Prod.snd

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

Function.Injective Prod.fst

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

Function.Injective Prod.snd

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def Prod.swap {α : Type u_1} {β : Type u_2} :

α × β → β × α

Swap the factors of a product. swap (a, b) = (b, a)

Equations

p.swap = (p.2, p.1)

Instances For

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@[simp]

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

x.swap.swap = x

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@[simp]

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

p.swap.1 = p.2

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@[simp]

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

p.swap.2 = p.1

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@[simp]

theorem Prod.swap_prod_mk {α : Type u_1} {β : Type u_2} {a : α} {b : β} :

(a, b).swap = (b, a)

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@[simp]

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

Prod.swap ∘ Prod.swap = id

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@[simp]

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

Function.LeftInverse Prod.swap Prod.swap

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@[simp]

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

Function.RightInverse Prod.swap Prod.swap

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

Function.Injective Prod.swap

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

Function.Surjective Prod.swap

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

Function.Bijective Prod.swap

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@[simp]

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

p.swap = q.swap ↔ p = q

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

Prod.map f g ∘ Prod.swap = Prod.swap ∘ Prod.map g f

For two functions f and g, the composition of Prod.map f g with Prod.swap is equal to the composition of Prod.swap with Prod.map g f.

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

Function.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 : α × β} {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 : α × β} {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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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)

Equations

Prod.Lex.decidable r s x✝ 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) [IsRefl α r] (x : α × β) :

Prod.Lex r s x x

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

IsRefl (α × β) (Prod.Lex r s)

Equations

⋯ = ⋯

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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) [IsRefl β s] (x : α × β) :

Prod.Lex r s x x

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instance Prod.instIsReflLex_1 {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsRefl β s] :

IsRefl (α × β) (Prod.Lex r s)

Equations

⋯ = ⋯

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

IsIrrefl (α × β) (Prod.Lex r s)

Equations

⋯ = ⋯

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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 : α × β} {y : α × β} {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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⋯ = ⋯

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instance Prod.instIsAntisymmLexOfIsStrictOrder {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsStrictOrder α r] [IsAntisymm β s] :

IsAntisymm (α × β) (Prod.Lex r s)

Equations

⋯ = ⋯

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

IsTotal (α × β) (Prod.Lex r s)

Equations

⋯ = ⋯

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instance Prod.isTotal_right {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrichotomous α r] [IsTotal β s] :

IsTotal (α × β) (Prod.Lex r s)

Equations

⋯ = ⋯

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

IsTrichotomous (α × β) (Prod.Lex r s)

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⋯ = ⋯

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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 : Function.Injective f) (hg : Function.Injective g) :

Function.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 : Function.Surjective f) (hg : Function.Surjective g) :

Function.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 : Function.Bijective f) (hg : Function.Bijective g) :

Function.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 : Function.LeftInverse f₁ f₂) (hg : Function.LeftInverse g₁ g₂) :

Function.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} :

Function.RightInverse f₁ f₂ → Function.RightInverse g₁ g₂ → Function.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} :

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

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@[deprecated Function.Injective.prodMap]

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

Function.Injective (Prod.map f g)

Alias of Function.Injective.prodMap.

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@[deprecated Function.Surjective.prodMap]

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

Function.Surjective (Prod.map f g)

Alias of Function.Surjective.prodMap.

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@[deprecated Function.Bijective.prodMap]

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

Function.Bijective (Prod.map f g)

Alias of Function.Bijective.prodMap.

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@[deprecated Function.LeftInverse.prodMap]

theorem Function.LeftInverse.Prod_map {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 : Function.LeftInverse f₁ f₂) (hg : Function.LeftInverse g₁ g₂) :

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

Alias of Function.LeftInverse.prodMap.

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@[deprecated Function.RightInverse.prodMap]

theorem Function.RightInverse.Prod_map {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} :

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

Alias of Function.RightInverse.prodMap.

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@[deprecated Function.Involutive.prodMap]

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

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

Alias of Function.Involutive.prodMap.

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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 (Prod.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 (Prod.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 (Prod.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 (Prod.map f₁ g₁) (Prod.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} :