norm_eq_of_add_equiv_zero — Mathlib

English

If f.norm ≠ g.norm, then (f + g).norm = max(f.norm, g.norm).

Русский

Если нормы различны, то норма суммы равна максимуму норм.

LaTeX

$$f.norm ≠ g.norm \Rightarrow (f + g).norm = \max\{f.norm, g.norm\}$$

Lean4

theorem norm_eq_of_add_equiv_zero {f g : PadicSeq p} (h : f + g ≈ 0) : f.norm = g.norm :=
  by
  have : LimZero (f + g - 0) := h
  have : f ≈ -g := show LimZero (f - -g) by simpa only [sub_zero, sub_neg_eq_add]
  have : f.norm = (-g).norm := norm_equiv this
  simpa only [norm_neg] using this

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