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.

@[simp]
theorem Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : αγ) (g : βδ) (p : α × β) :
Prod.map f g p = (f p.1, g p.2)
@[simp]
theorem Prod.mk.eta {α : Type u_1} {β : Type u_2} {p : α × β} :
(p.1, p.2) = p
@[simp]
theorem Prod.forall {α : Type u_1} {β : Type u_2} {p : α × βProp} :
(∀ (x : α × β), p x) ∀ (a : α) (b : β), p (a, b)
@[simp]
theorem Prod.exists {α : Type u_1} {β : Type u_2} {p : α × βProp} :
(∃ (x : α × β), p x) ∃ (a : α), ∃ (b : β), p (a, b)
theorem Prod.forall' {α : Type u_1} {β : Type u_2} {p : αβProp} :
(∀ (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 : αβProp} :
(∃ (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 = id
@[simp]
theorem Prod.fst_comp_mk {α : Type u_1} {β : Type u_2} (x : α) :
Prod.fst =
@[simp]
theorem Prod.map_mk {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : αγ) (g : βδ) (a : α) (b : β) :
Prod.map f g (a, b) = (f a, g b)
theorem Prod.map_fst {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : αγ) (g : βδ) (p : α × β) :
(Prod.map f g p).1 = f p.1
theorem Prod.map_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : αγ) (g : βδ) (p : α × β) :
(Prod.map f g p).2 = g p.2
theorem Prod.map_fst' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : αγ) (g : βδ) :
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 : αγ) (g : βδ) :
Prod.snd Prod.map f g = g Prod.snd
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 : αβ) (f' : γδ) (g : βε) (g' : δζ) :
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.

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 : αβ) (f' : γδ) (g : βε) (g' : δζ) (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.

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
theorem Prod.mk.inj_left {α : Type u_5} {β : Type u_6} (a : α) :
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 : β} {b₂ : β} :
(a, b₁) = (a, b₂) b₁ = b₂
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
theorem Prod.ext_iff {α : Type u_1} {β : Type u_2} {p : α × β} {q : α × β} :
p = q p.1 = q.1 p.2 = q.2
theorem Prod.map_def {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : αγ} {g : βδ} :
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_id {α : Type u_1} {β : Type u_2} :
Prod.map id id = id
theorem Prod.fst_surjective {α : Type u_1} {β : Type u_2} [h : ] :
theorem Prod.snd_surjective {α : Type u_1} {β : Type u_2} [h : ] :
theorem Prod.fst_injective {α : Type u_1} {β : Type u_2} [] :
theorem Prod.snd_injective {α : Type u_1} {β : Type u_2} [] :
def Prod.swap {α : Type u_1} {β : Type u_2} :
α × ββ × α

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

Equations
• p.swap = (p.2, p.1)
Instances For
@[simp]
theorem Prod.swap_swap {α : Type u_1} {β : Type u_2} (x : α × β) :
x.swap.swap = x
@[simp]
theorem Prod.fst_swap {α : Type u_1} {β : Type u_2} {p : α × β} :
p.swap.1 = p.2
@[simp]
theorem Prod.snd_swap {α : Type u_1} {β : Type u_2} {p : α × β} :
p.swap.2 = p.1
@[simp]
theorem Prod.swap_prod_mk {α : Type u_1} {β : Type u_2} {a : α} {b : β} :
(a, b).swap = (b, a)
@[simp]
theorem Prod.swap_swap_eq {α : Type u_1} {β : Type u_2} :
Prod.swap Prod.swap = id
@[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} :
theorem Prod.swap_surjective {α : Type u_1} {β : Type u_2} :
theorem Prod.swap_bijective {α : Type u_1} {β : Type u_2} :
@[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
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
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 : ααProp} {s : ββProp} {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
instance Prod.Lex.decidable {α : Type u_1} {β : Type u_2} [] (r : ααProp) (s : ββProp) [] [] :
Equations
theorem Prod.Lex.refl_left {α : Type u_1} {β : Type u_2} (r : ααProp) (s : ββProp) [IsRefl α r] (x : α × β) :
Prod.Lex r s x x
instance Prod.instIsReflLex {α : Type u_1} {β : Type u_2} {r : ααProp} {s : ββProp} [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 : ααProp) (s : ββProp) [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 : ααProp} {s : ββProp} [IsRefl β s] :
IsRefl (α × β) (Prod.Lex r s)
Equations
• =
instance Prod.isIrrefl {α : Type u_1} {β : Type u_2} {r : ααProp} {s : ββProp} [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 : ααProp} {s : ββProp} [IsTrans α r] [IsTrans β s] {x : α × β} {y : α × β} {z : α × β} :
Prod.Lex r s x yProd.Lex r s y zProd.Lex r s x z
instance Prod.instIsTransLex {α : Type u_1} {β : Type u_2} {r : ααProp} {s : ββProp} [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 : ααProp} {s : ββProp} [] [] :
IsAntisymm (α × β) (Prod.Lex r s)
Equations
• =
instance Prod.isTotal_left {α : Type u_1} {β : Type u_2} {r : ααProp} {s : ββProp} [IsTotal α r] :
IsTotal (α × β) (Prod.Lex r s)
Equations
• =
instance Prod.isTotal_right {α : Type u_1} {β : Type u_2} {r : ααProp} {s : ββProp} [] [IsTotal β s] :
IsTotal (α × β) (Prod.Lex r s)
Equations
• =
instance Prod.IsTrichotomous {α : Type u_1} {β : Type u_2} {r : ααProp} {s : ββProp} [] [] :
IsTrichotomous (α × β) (Prod.Lex r s)
Equations
• =
theorem Function.Injective.Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : αγ} {g : βδ} (hf : ) (hg : ) :
theorem Function.Surjective.Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : αγ} {g : βδ} (hf : ) (hg : ) :
theorem Function.Bijective.Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : αγ} {g : βδ} (hf : ) (hg : ) :
theorem Function.LeftInverse.Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f₁ : αβ} {g₁ : γδ} {f₂ : βα} {g₂ : δγ} (hf : ) (hg : ) :
Function.LeftInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂)
theorem Function.RightInverse.Prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f₁ : αβ} {g₁ : γδ} {f₂ : βα} {g₂ : δγ} :
Function.RightInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂)
theorem Function.Involutive.Prod_map {α : Type u_1} {β : Type u_2} {f : αα} {g : ββ} :
@[simp]
theorem Prod.map_injective {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [] [] {f : αγ} {g : βδ} :
@[simp]
theorem Prod.map_surjective {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [] [] {f : αγ} {g : βδ} :
@[simp]
theorem Prod.map_bijective {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [] [] {f : αγ} {g : βδ} :
@[simp]
theorem Prod.map_leftInverse {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [] [] {f₁ : αβ} {g₁ : γδ} {f₂ : βα} {g₂ : δγ} :
Function.LeftInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂)
@[simp]
theorem Prod.map_rightInverse {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [] [] {f₁ : αβ} {g₁ : γδ} {f₂ : βα} {g₂ : δγ} :
Function.RightInverse (Prod.map f₁ g₁) (Prod.map f₂ g₂)
@[simp]
theorem Prod.map_involutive {α : Type u_1} {β : Type u_2} [] [] {f : αα} {g : ββ} :

Delaborator for Prod.fst x as x.1.

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

Delaborator for Prod.snd x as x.2.

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