Metamath Proof Explorer


Theorem cdlemeulpq

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 5-Dec-2012)

Ref Expression
Hypotheses cdleme0.l ˙ = K
cdleme0.j ˙ = join K
cdleme0.m ˙ = meet K
cdleme0.a A = Atoms K
cdleme0.h H = LHyp K
cdleme0.u U = P ˙ Q ˙ W
Assertion cdlemeulpq K HL W H P A Q A U ˙ P ˙ Q

Proof

Step Hyp Ref Expression
1 cdleme0.l ˙ = K
2 cdleme0.j ˙ = join K
3 cdleme0.m ˙ = meet K
4 cdleme0.a A = Atoms K
5 cdleme0.h H = LHyp K
6 cdleme0.u U = P ˙ Q ˙ W
7 simpll K HL W H P A Q A K HL
8 7 hllatd K HL W H P A Q A K Lat
9 simprl K HL W H P A Q A P A
10 simprr K HL W H P A Q A Q A
11 eqid Base K = Base K
12 11 2 4 hlatjcl K HL P A Q A P ˙ Q Base K
13 7 9 10 12 syl3anc K HL W H P A Q A P ˙ Q Base K
14 11 5 lhpbase W H W Base K
15 14 ad2antlr K HL W H P A Q A W Base K
16 11 1 3 latmle1 K Lat P ˙ Q Base K W Base K P ˙ Q ˙ W ˙ P ˙ Q
17 8 13 15 16 syl3anc K HL W H P A Q A P ˙ Q ˙ W ˙ P ˙ Q
18 6 17 eqbrtrid K HL W H P A Q A U ˙ P ˙ Q