Extra facts about Prod

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

@[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.

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

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

@[simp]

theorem Prod.mk.eta {α : Type u_1} {β : Type u_2} {p : α × β} : (p.1, p.2) = p

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

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

@[simp]

theorem Prod.snd_comp_mk {α : Type u_1} {β : Type u_2} (x : α) : Prod.snd ∘ Prod.mk x = id

@[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

@[deprecated Prod.map_apply]

theorem Prod.map_mk {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.

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)

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

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

theorem Prod.mk.inj_iff {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂

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

theorem Prod.mk.inj_right {α : Type u_5} {β : Type u_6} (b : β) : Function.Injective fun (a : α) => (a, b)

theorem Prod.mk_inj_left {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} : (a, b₁) = (a, b₂) ↔ b₁ = b₂

theorem Prod.mk_inj_right {α : Type u_1} {β : Type u_2} {a₁ a₂ : α} {b : β} : (a₁, b) = (a₂, b) ↔ a₁ = a₂

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)

theorem Prod.id_prod {α : Type u_1} {β : Type u_2} : (fun (p : α × β) => (p.1, p.2)) = id

@[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]

@[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.

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

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

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

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

@[simp]

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

@[simp]

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

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

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

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

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)

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

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)

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)

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

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 ⋯

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

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

• ⋯ = ⋯

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

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

• ⋯ = ⋯

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

• ⋯ = ⋯

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

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)

Equations

• ⋯ = ⋯

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

• ⋯ = ⋯

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

• ⋯ = ⋯

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

• ⋯ = ⋯

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)

Equations

• ⋯ = ⋯

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)

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)

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)

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)

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)

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)

@[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.

@[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.

@[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.

@[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.

@[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.

@[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.

@[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

@[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

@[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

@[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

@[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} : Function.RightInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂) ↔ Function.RightInverse f₁ f₂ ∧ Function.RightInverse g₁ g₂

@[simp]

theorem Prod.map_involutive {α : Type u_1} {β : Type u_2} [Nonempty α] [Nonempty β] {f : α → α} {g : β → β} : Function.Involutive (Prod.map f g) ↔ Function.Involutive f ∧ Function.Involutive g

def Prod.delabProdFst : Lean.PrettyPrinter.Delaborator.Delab

Delaborator for Prod.fst x as x.1.

Equations

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

def Prod.delabProdSnd : Lean.PrettyPrinter.Delaborator.Delab

Delaborator for Prod.snd x as x.2.

Equations

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