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[Merged by Bors] - feat: Add Nat.card_sum #6081

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[Merged by Bors] - feat: Add Nat.card_sum #6081

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theorem
tb65536 Jul 23, 2023
a9f412b
simp
tb65536 Jul 24, 2023
9fffe36
Update Mathlib/SetTheory/Cardinal/Finite.lean
tb65536 Jul 24, 2023
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6 changes: 6 additions & 0 deletions Mathlib/SetTheory/Cardinal/Finite.lean
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Expand Up @@ -95,6 +95,12 @@ theorem card_eq_two_iff' (x : α) : Nat.card α = 2 ↔ ∃! y, y ≠ x :=
theorem card_of_isEmpty [IsEmpty α] : Nat.card α = 0 := by simp
#align nat.card_of_is_empty Nat.card_of_isEmpty

@[simp]
theorem card_sum [Finite α] [Finite β] : Nat.card (α ⊕ β) = Nat.card α + Nat.card β := by
have := Fintype.ofFinite α
have := Fintype.ofFinite β
simp_rw [Nat.card_eq_fintype_card, Fintype.card_sum]

@[simp]
theorem card_prod (α β : Type _) : Nat.card (α × β) = Nat.card α * Nat.card β := by
simp only [Nat.card, mk_prod, toNat_mul, toNat_lift]
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