Theorems · Theorem · group theory

Finset.sum_bij

∀ {ι : Type u_1} {κ : Type u_2} {M : Type u_3} [inst : AddCommMonoid M] {s : Finset ι} {t : Finset κ} {f : ι → M}
  {g : κ → M} (i : (a : ι) → a ∈ s → κ),
  (∀ (a : ι) (ha : a ∈ s), i a ha ∈ t) →
    (∀ (a₁ : ι) (ha₁ : a₁ ∈ s) (a₂ : ι) (ha₂ : a₂ ∈ s), i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂) →
      (∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), i a ha = b) →
        (∀ (a : ι) (ha : a ∈ s), f a = g (i a ha)) → ∑ x ∈ s, f x = ∑ x ∈ t, g x

Reorder a sum. The difference with Finset.sum_bij' is that the bijection is specified as a surjective injection, rather than by an inverse function. The difference with Finset.sum_nbij is that the bijection is allowed to use membership of the domain of the sum, rather than being a non-dependent function.

Defined in

Mathlib.Algebra.BigOperators.Group.Finset.Defs

Cited by

16 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoid