gi — Mathlib · SciLib

English

If e: α ≃o β is an order isomorphism, then IsCofinal s → IsCofinal (e '' s).

Русский

Если e: α ≃o β — упорядоченное изоморфизм, то из Cofinal s следует Cofinal (e''s).

LaTeX

$$$\text{IsCofinal}(s) \rightarrow \text{IsCofinal}(e''s)$$$

Lean4

/-- The set of closed elements has a Galois insertion to the underlying type. -/
def gi [PartialOrder α] (c : ClosureOperator α) : GaloisInsertion c.toCloseds (↑)
    where
  choice x hx := ⟨x, isClosed_iff_closure_le.2 hx⟩
  gc _ y := y.2.closure_le_iff
  le_l_u _ := c.le_closure _
  choice_eq x hx := le_antisymm (c.le_closure x) hx

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