Lemma isaposetmorphism_pointwise_lub {P : Poset} {I} {f : I -> P -> P} (f_monot : forall i, isaposetmorphism (f i)) {g} (g_pw_lub : is_least_upper_bound_pointwise f g) : isaposetmorphism g. Proof. intros x y le_x_y. apply (least_upper_bound_univ (make_least_upper_bound _ (g_pw_lub _))). intros i. eapply istrans_posetRelation. 2: { apply (least_upper_bound_is_upper_bound (make_least_upper_bound _ (g_pw_lub _))). } apply f_monot; assumption. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.OrderedSets.WellOrderedSets. Require Import UniMath.OrderTheory.DCPOs.AlternativeDefinitions.Dcpo. Require Import UniMath.OrderTheory.Posets.Subposets.