coe_iSup — Mathlib

English

For nonempty ι, the coercion commutes with iSup: the coe of iSup equals iSup of coerced elements.

Русский

Для непустого ι отображение коe commuting с iSup: коe(iSup i) = iSup i (коe S(i)).

LaTeX

$$$$ \uparrow(\operatorname{iSup} S) = \operatorname{iSup}\big( \uparrow( S(i) ) \big). $$$$

Lean4

theorem coe_iSup [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b) [Nonempty ι] {S : ι → Set.Icc a b} :
    have : Fact (a ≤ b) := ⟨h⟩
    ↑(iSup S) = (⨆ i, S i : α) :=
  (Set.Icc.coe_sSup h (range_nonempty S)).trans (congrArg sSup (range_comp Subtype.val S).symm)

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