English
Under a Galois connection, the image of a least upper bound is a least upper bound of the image, i.e., l preserves lubs along sets.
Русский
При Гало связи образ наименьшей верхней границы сохраняет верхнюю границу образа, то есть l сохраняет lub по множествам.
LaTeX
$$$$\\text{IsLUB}(l''s) (l a) \\text{ from IsLUB}(s,a)$$$$
Lean4
theorem isLUB_l_image {s : Set α} {a : α} (h : IsLUB s a) : IsLUB (l '' s) (l a) :=
⟨gc.monotone_l.mem_upperBounds_image h.left, fun b hb => gc.l_le <| h.right <| by rwa [gc.upperBounds_l_image] at hb⟩