piCongrRight — Mathlib

English

The image of g under smulLeft, applied to x, equals the action of g on x: smulLeft(g)(x) = g·x.

Русский

Изображение g под smulLeft, примененное к x, равно действию g на x: smulLeft(g)(x) = g·x.

LaTeX

$$$smulLeft\\_apply(g,x) : smulLeft(g)(x) = g\\cdot x$$$

Lean4

/-- Combine a family of continuous linear equivalences into a continuous linear equivalence of
pi-types. -/
def piCongrRight : ((i : ι) → M i) ≃L[R₁] (i : ι) → N i :=
  {
    LinearEquiv.piCongrRight fun i ↦
      f
        i with
    continuous_toFun := by exact continuous_pi fun i ↦ (f i).continuous_toFun.comp (continuous_apply i)
    continuous_invFun := by exact continuous_pi fun i => (f i).continuous_invFun.comp (continuous_apply i) }

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