# Documentation

Mathlib.Data.Fintype.Prod

# fintype instance for the product of two fintypes. #

theorem Set.toFinset_prod {α : Type u_1} {β : Type u_2} (s : Set α) (t : Set β) [inst : Fintype s] [inst : Fintype t] [inst : Fintype ↑(s ×ˢ t)] :
theorem Set.toFinset_off_diag {α : Type u_1} {s : Set α} [inst : ] [inst : Fintype s] [inst : Fintype ↑()] :
instance instFintypeProd (α : Type u_1) (β : Type u_2) [inst : ] [inst : ] :
Fintype (α × β)
Equations
• = { elems := Finset.univ ×ᶠ Finset.univ, complete := (_ : ∀ (x : α × β), x Finset.univ ×ᶠ Finset.univ) }
@[simp]
theorem Finset.univ_product_univ {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] :
Finset.univ ×ᶠ Finset.univ = Finset.univ
@[simp]
theorem Fintype.card_prod (α : Type u_1) (β : Type u_2) [inst : ] [inst : ] :
Fintype.card (α × β) =
@[simp]
theorem infinite_prod {α : Type u_2} {β : Type u_1} :
Infinite (α × β)
instance Pi.infinite_of_left {ι : Sort u_1} {π : ιType u_2} [inst : ∀ (i : ι), Nontrivial (π i)] [inst : ] :
Infinite ((i : ι) → π i)
Equations
theorem Pi.infinite_of_exists_right {ι : Type u_1} {π : ιType u_2} (i : ι) [inst : Infinite (π i)] [inst : ∀ (i : ι), Nonempty (π i)] :
Infinite ((i : ι) → π i)

If at least one π i is infinite and the rest nonempty, the pi type of all π is infinite.

instance Pi.infinite_of_right {ι : Type u_1} {π : ιType u_2} [inst : ∀ (i : ι), Infinite (π i)] [inst : ] :
Infinite ((i : ι) → π i)

See Pi.infinite_of_exists_right for the case that only one π i is infinite.

Equations
instance Function.infinite_of_left {ι : Sort u_1} {π : Type u_2} [inst : ] [inst : ] :
Infinite (ιπ)

Non-dependent version of Pi.infinite_of_left.

Equations
instance Function.infinite_of_right {ι : Type u_1} {π : Type u_2} [inst : ] [inst : ] :
Infinite (ιπ)

Non-dependent version of Pi.infinite_of_exists_right and Pi.infinite_of_right.

Equations