hull_inf — Mathlib

English

hull_inf: hull(T, a ⊓ b) = hull(T,a) ∪ hull(T,b) under InfPrime condition.

Русский

hull_inf: hull(T, a ⊓ b) = hull(T,a) ∪ hull(T,b) при условии InfPrime.

LaTeX

$$$\\operatorname{hull}(T,a\\wedge b) = \\operatorname{hull}(T,a) \\cup \\operatorname{hull}(T,b)$ given InfPrime(p) for p ∈ T.$$

Lean4

theorem hull_inf (hT : ∀ p ∈ T, InfPrime p) (a b : α) : hull T (a ⊓ b) = hull T a ∪ hull T b :=
  by
  ext p
  constructor <;> intro h
  · exact (hT p p.2).2 h
  · rcases h with (h1 | h3)
    · exact inf_le_of_left_le h1
    · exact inf_le_of_right_le h3

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