Theorems · Theorem · group theory

Finset.prod_bij_ne_one

∀ {ι : Type u_1} {κ : Type u_2} {M : Type u_4} [inst : CommMonoid M] {s : Finset ι} {t : Finset κ} {f : ι → M}
  {g : κ → M} (i : (a : ι) → a ∈ s → f a ≠ 1 → κ),
  (∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t) →
    (∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),
        i a₁ h₁₁ h₁₂ = i a₂ h₂₁ h₂₂ → a₁ = a₂) →
      (∀ b ∈ t, g b ≠ 1 → ∃ a, ∃ (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ = b) →
        (∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), f a = g (i a h₁ h₂)) → ∏ x ∈ s, f x = ∏ x ∈ t, g x

Defined in

Mathlib.Algebra.BigOperators.Group.Finset.Basic

Cited by

0 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoid