smul_prod_smul — Mathlib

English

Another extensionality lemma for product multilinear maps: equality on fst and snd implies equality of maps.

Русский

Еще одно лемма экстенсиональности для отображений в произведение: равенство по fst и snd влечет равенство карт.

LaTeX

$$$\text{fst}\circ f = \text{fst}\circ g \quad \text{and} \quad \text{snd}\circ f = \text{snd}\circ g \Rightarrow f=g$$$

Lean4

theorem smul_prod_smul {S : Type*} [Monoid S] [DistribMulAction S M₂] [DistribMulAction S M₃] [ContinuousConstSMul S M₂]
    [SMulCommClass R S M₂] [ContinuousConstSMul S M₃] [SMulCommClass R S M₃] (c : S)
    (f : ContinuousMultilinearMap R M₁ M₂) (g : ContinuousMultilinearMap R M₁ M₃) :
    (c • f).prod (c • g) = c • f.prod g :=
  rfl

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