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