prod_erase_attach — Mathlib

English

The cardinality of the sum of a finite family of multisets equals the sum of their cardinalities: card(∑_{i∈s} f(i)) = ∑_{i∈s} card(f(i)).

Русский

Кардинал совокупности сумм множества множества равен сумме их кардиналов: card(∑ f(i)) = ∑ card(f(i)).

LaTeX

$$$$\operatorname{card}\left(\sum_{i \in s} f(i)\right) = \sum_{i \in s} \operatorname{card}(f(i)).$$$$

Lean4

@[to_additive]
theorem prod_erase_attach [DecidableEq ι] {s : Finset ι} (f : ι → M) (i : ↑s) :
    ∏ j ∈ s.attach.erase i, f ↑j = ∏ j ∈ s.erase ↑i, f j :=
  by
  rw [← Function.Embedding.coe_subtype, ← prod_map]
  simp [attach_map_val]

Похожие утверждения