Documentation

Mathlib.Data.Nat.Cast.Prod

The product of two AddMonoidWithOnes. #

source
instance Prod.instAddMonoidWithOne {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] :
AddMonoidWithOne (α × β)
Equations
  • Prod.instAddMonoidWithOne = let __src := Prod.instAddMonoid; let __src_1 := Prod.instOne; AddMonoidWithOne.mk ⋯ ⋯
source
@[simp]
theorem Prod.fst_natCast {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] (n : ℕ) :
(↑n).1 = ↑n
source
@[simp]
theorem Prod.fst_ofNat {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] (n : ℕ) [Nat.AtLeastTwo n] :
(OfNat.ofNat n).1 = OfNat.ofNat n
source
@[simp]
theorem Prod.snd_natCast {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] (n : ℕ) :
(↑n).2 = ↑n
source
@[simp]
theorem Prod.snd_ofNat {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] (n : ℕ) [Nat.AtLeastTwo n] :
(OfNat.ofNat n).2 = OfNat.ofNat n