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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 
|- .<_ = ( 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 )
Assertion cdlemeulpq 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )

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 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> K e. HL )
8 7 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> K e. Lat )
9 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> P e. A )
10 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> Q e. A )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
13 7 9 10 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
14 11 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
15 14 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> W e. ( Base ` K ) )
16 11 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
17 8 13 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
18 6 17 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )