support_sup — Mathlib

English

The support of the pointwise supremum equals the union of supports: (f ⊔ g).support = f.support ∪ g.support.

Русский

Опора точечного верхнего也 предельного равна объединению опор: supp(f ⊔ g) = supp(f) ∪ supp(g).

LaTeX

$$$ (f \\sup g).\\operatorname{support} = f.\\operatorname{support} \\cup g.\\operatorname{support} $$$

Lean4

@[simp]
theorem support_sup [DecidableEq ι] (f g : ι →₀ α) : (f ⊔ g).support = f.support ∪ g.support :=
  by
  ext
  simp only [mem_support_iff, Ne, sup_apply, ← nonpos_iff_eq_zero, sup_le_iff, mem_union, not_and_or]

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