exists_eq_u — Mathlib

English

For z ∈ α and y ∈ β, u y = z iff ∀ x ∈ α, x ≤ z ↔ l x ≤ y.

Русский

Для z ∈ α и y ∈ β: u y = z эквивалентно ∀ x ∈ α, x ≤ z ↔ l x ≤ y.

LaTeX

$$$u(y) = z \iff \forall x:\alpha, x \le z \;\iff\; l(x) \le y$$$

Lean4

/-- If there exists a `b` such that `a = u a`, then `b = l a` is one such element. -/
theorem exists_eq_u (a : α) : (∃ b : β, a = u b) ↔ a = u (l a) :=
  ⟨fun ⟨_, hS⟩ => hS.symm ▸ (gc.u_l_u_eq_u _).symm, fun HI => ⟨_, HI⟩⟩

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