Theorems · Theorem · real analysis

Monovary.sum_smul_comp_perm_eq_sum_smul_iff

∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Semiring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]
  [ExistsAddOfLE α] [inst_4 : AddCommMonoid β] [inst_5 : LinearOrder β] [IsOrderedCancelAddMonoid β]
  [inst_7 : Module α β] [PosSMulStrictMono α β] {σ : Equiv.Perm ι} {f : ι → α} {g : ι → β} [inst_9 : Fintype ι],
  Monovary f g → (∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ Monovary f (g ∘ ⇑σ))

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

Defined in

Mathlib.Algebra.Order.Rearrangement

Cited by

1 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringLinearOrderIsStrictOrderedRingExistsAddOfLEAddCommMonoidLinearOrderIsOrderedCancelAddMonoidModulePosSMulStrictMonoFintype