This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdleme0aa

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 14-Jun-2012)

Ref Expression
Hypotheses cdleme0.l 
|- .<_ = ( le ` 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 )
cdleme0.b 
|- B = ( Base ` K )
Assertion cdleme0aa 
|- ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. B )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` 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 cdleme0.b 
 |-  B = ( Base ` K )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> K e. HL )
9 8 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> K e. Lat )
10 7 4 atbase 
 |-  ( P e. A -> P e. B )
11 10 3ad2ant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> P e. B )
12 7 4 atbase 
 |-  ( Q e. A -> Q e. B )
13 12 3ad2ant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> Q e. B )
14 7 2 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
15 9 11 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B )
16 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> W e. H )
17 7 5 lhpbase 
 |-  ( W e. H -> W e. B )
18 16 17 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> W e. B )
19 7 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ W e. B ) -> ( ( P .\/ Q ) ./\ W ) e. B )
20 9 15 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> ( ( P .\/ Q ) ./\ W ) e. B )
21 6 20 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. B )