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[Merged by Bors] - feat: Unions and intersections indexed by Finset.sigma #10851

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[Merged by Bors] - feat: Unions and intersections indexed by Finset.sigma #10851

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212d06e
feat: Unions and intersections indexed by `Finset.sigma`
YaelDillies Feb 22, 2024
124b3c1
dedup namespace
YaelDillies Feb 22, 2024
e94fd91
use `Π`
YaelDillies Feb 28, 2024
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2 changes: 2 additions & 0 deletions Mathlib/Data/Fintype/Card.lean
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Expand Up @@ -538,6 +538,8 @@ theorem card_eq_zero_iff : card α = 0 ↔ IsEmpty α := by
card_eq_zero_iff.2 ‹_›
#align fintype.card_eq_zero Fintype.card_eq_zero

alias card_of_isEmpty := card_eq_zero

theorem card_eq_one_iff_nonempty_unique : card α = 1 ↔ Nonempty (Unique α) :=
⟨fun h =>
let ⟨d, h⟩ := Fintype.card_eq_one_iff.mp h
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14 changes: 13 additions & 1 deletion Mathlib/Data/Fintype/Sigma.lean
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Expand Up @@ -19,7 +19,7 @@ open Nat

universe u v

variable {α β γ : Type*}
variable {ι α β γ : Type*} {κ : ι → Type*} [∀ i, Fintype (κ i)]

open Finset Function

Expand All @@ -36,3 +36,15 @@ instance PSigma.fintype {α : Type*} {β : α → Type*} [Fintype α] [∀ a, Fi
Fintype (Σ'a, β a) :=
Fintype.ofEquiv _ (Equiv.psigmaEquivSigma _).symm
#align psigma.fintype PSigma.fintype

lemma Set.biUnion_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :
⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋃ i ∈ s, ⋃ j, f ⟨i, j⟩ := by aesop

lemma Set.biUnion_finsetSigma_univ' (s : Finset ι) (f : ∀ i, κ i → Set α) :
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I am not a big fan of the notatiion f : ∀ i, κ i → Set α, but I guess PiNotation is not available here.

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It's available, actually, but I'm afraid using it is a lost cause. I can only find it in comments.

⋃ i ∈ s, ⋃ j, f i j = ⋃ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 := by aesop

lemma Set.biInter_finsetSigma_univ (s : Finset ι) (f : Sigma κ → Set α) :
⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij = ⋂ i ∈ s, ⋂ j, f ⟨i, j⟩ := by aesop

lemma Set.biInter_finsetSigma_univ' (s : Finset ι) (f : ∀ i, κ i → Set α) :
⋂ i ∈ s, ⋂ j, f i j = ⋂ ij ∈ s.sigma fun _ ↦ Finset.univ, f ij.1 ij.2 := by aesop