csSup_image2_eq_csInf_csInf — Mathlib

English

Under dual Galois connections, the two-argument image infimum equals a fixed infimum composition.

Русский

При двойственных Гало-соединениях инфимума образа двух аргументов равна фиксированной композиции инфимумов.

LaTeX

$$$(h_1,h_2): s.Nonempty \Rightarrow BddBelow s \Rightarrow t.Nonempty \Rightarrow BddAbove t \Rightarrow sInf (image2 u s t) = u (sInf s) (sInf t)$$$

Lean4

theorem csSup_image2_eq_csInf_csInf (h₁ : ∀ b, GaloisConnection (swap l b ∘ ofDual) (toDual ∘ u₁ b))
    (h₂ : ∀ a, GaloisConnection (l a ∘ ofDual) (toDual ∘ u₂ a)) :
    s.Nonempty → BddBelow s → t.Nonempty → BddBelow t → sSup (image2 l s t) = l (sInf s) (sInf t) :=
  csSup_image2_eq_csSup_csSup (α := αᵒᵈ) (β := βᵒᵈ) h₁ h₂

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