sSup_apply — Mathlib

English

If l ∈ c is such that hl ∈ c, then (sSup c)(x) = l(x) for appropriate x in domain.

Русский

Если l ∈ c и hl соответствует l, то (sup c)(x) = l(x) на соответствующем x.

LaTeX

$$$$ (\\mathrm{sSup}(c))\\;\\text{toFun'}\\langle x, \\cdots \\rangle = l(x) $$$$

Lean4

protected theorem sSup_apply {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (· ≤ ·) c) {l : E →ₗ.[R] F} (hl : l ∈ c)
    (x : l.domain) : (LinearPMap.sSup c hc) ⟨x, (LinearPMap.le_sSup hc hl).1 x.2⟩ = l x :=
  by
  symm
  apply (Classical.choose_spec (sSup_aux c hc) hl).2
  rfl

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