Theorems · Theorem · group theory

Finset.prod_involution

∀ {ι : Type u_1} {M : Type u_4} {s : Finset ι} [inst : CommMonoid M] {f : ι → M} (g : (a : ι) → a ∈ s → ι),
  (∀ (a : ι) (ha : a ∈ s), f a * f (g a ha) = 1) →
    (∀ (a : ι) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) →
      ∀ (g_mem : ∀ (a : ι) (ha : a ∈ s), g a ha ∈ s), (∀ (a : ι) (ha : a ∈ s), g (g a ha) ⋯ = a) → ∏ x ∈ s, f x = 1

The difference with Finset.prod_ninvolution is that the involution is allowed to use membership of the domain of the product, rather than being a non-dependent function.

Defined in

Mathlib.Algebra.BigOperators.Group.Finset.Basic

Cited by

2 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoid