element_smul_restrictScalars — Mathlib

English

map distributes over suprema of sets: map f (sSup s) = iSup I∈s, map f I.

Русский

образ распределяется по верхним пределам множества: map f (sSup s) = ⨆ I∈s, map f I.

LaTeX

$$$\\operatorname{map} f\\big(\\operatorname{Sups} s\\big) = \\big\\langle \\text{iSup over } I\\in s\\big\\rangle (\\operatorname{map} f I)$$$

Lean4

theorem _root_.element_smul_restrictScalars {R S M} [CommSemiring R] [CommSemiring S] [Algebra R S] [AddCommMonoid M]
    [Module R M] [Module S M] [IsScalarTower R S M] (r : R) (N : Submodule S M) :
    (algebraMap R S r • N).restrictScalars R = r • N.restrictScalars R :=
  SetLike.coe_injective (congrArg (· '' _) (funext (algebraMap_smul S r)))

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