lt_ofDual_iff — Mathlib

English

For x ∈ WithBot αᵒᵈ and y ∈ WithTop α, x < ToDual y is equivalent to y large too in the dual sense: y < OfDual x.

Русский

Для x ∈ WithBot αᵒᵈ и y ∈ WithTop α, x < toDual(y) эквивалентно y < ofDual(x) в двойственном порядке.

LaTeX

$$$ x < \mathrm{toDual}(y) \iff y < x^{\mathrm{toDual}}. $$$$

Lean4

theorem lt_ofDual_iff {x : WithBot α} {y : WithTop αᵒᵈ} : x < WithTop.ofDual y ↔ y < WithBot.toDual x := by
  cases x <;> cases y <;> simp [lt_toDual]

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