projectiveSeminormAux_smul — Mathlib

English

The projective seminorm equals the infimum of the projectiveSeminormAux over all lifts of x: projectiveSeminorm x = inf_p (projectiveSeminormAux p.1).

Русский

Проективная семиномормая равна инфимума по lifts x: projectiveSeminorm x = inf_p (projectiveSeminormAux p.1).

LaTeX

$$$ \\mathrm{projectiveSeminorm}(x) = \\inf_{p} \\mathrm{projectiveSeminormAux}(p.1) $$$

Lean4

theorem projectiveSeminormAux_smul (p : FreeAddMonoid (𝕜 × Π i, E i)) (a : 𝕜) :
    projectiveSeminormAux (p.map (fun (y : 𝕜 × Π i, E i) ↦ (a * y.1, y.2))) = ‖a‖ * projectiveSeminormAux p := by
  simp [projectiveSeminormAux, Function.comp_def, mul_assoc, List.sum_map_mul_left]

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