Theorems · Theorem · real analysis

AntivaryOn.sum_comp_perm_mul_eq_sum_mul_iff

∀ {ι : Type u_1} {α : Type u_2} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α]
  {s : Finset ι} {σ : Equiv.Perm ι} {f g : ι → α},
  AntivaryOn f g ↑s → {x | σ x ≠ x} ⊆ ↑s → (∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ AntivaryOn (f ∘ ⇑σ) g ↑s)

Equality case of the Rearrangement Inequality: Pointwise multiplication of f and g, which antivary together on s, is unchanged by a permutation if and only if f ∘ σ and g antivary together on s. Stated by permuting the entries of f.

Defined in

Mathlib.Algebra.Order.Rearrangement

Cited by

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLE