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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 e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> C e. 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 e. HL -> K e. Lat )
8 7 adantr 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> K e. Lat )
9 1 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. B )
10 9 3adant3r3 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> ( P .\/ S ) e. B )
11 simpr3 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> W e. H )
12 1 5 lhpbase 
 |-  ( W e. H -> W e. B )
13 11 12 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> W e. B )
14 1 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ S ) e. B /\ W e. B ) -> ( ( P .\/ S ) ./\ W ) e. B )
15 8 10 13 14 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> ( ( P .\/ S ) ./\ W ) e. B )
16 6 15 eqeltrid 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ W e. H ) ) -> C e. B )