Metamath Proof Explorer


Theorem cdleme9b

Description: Utility lemma for Lemma E in Crawley p. 113. (Contributed by NM, 9-Oct-2012)

Ref Expression
Hypotheses cdleme9b.b B = Base K
cdleme9b.j ˙ = join K
cdleme9b.m ˙ = meet K
cdleme9b.a A = Atoms K
cdleme9b.h H = LHyp K
cdleme9b.c C = P ˙ S ˙ W
Assertion cdleme9b K HL P A S A W H C B

Proof

Step Hyp Ref Expression
1 cdleme9b.b B = Base K
2 cdleme9b.j ˙ = join K
3 cdleme9b.m ˙ = meet K
4 cdleme9b.a A = Atoms K
5 cdleme9b.h H = LHyp K
6 cdleme9b.c C = P ˙ S ˙ W
7 hllat K HL K Lat
8 7 adantr K HL P A S A W H K Lat
9 1 2 4 hlatjcl K HL P A S A P ˙ S B
10 9 3adant3r3 K HL P A S A W H P ˙ S B
11 simpr3 K HL P A S A W H W H
12 1 5 lhpbase W H W B
13 11 12 syl K HL P A S A W H W B
14 1 3 latmcl K Lat P ˙ S B W B P ˙ S ˙ W B
15 8 10 13 14 syl3anc K HL P A S A W H P ˙ S ˙ W B
16 6 15 eqeltrid K HL P A S A W H C B