Metamath Proof Explorer

Theorem cdleme0cq

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 25-Apr-2013)

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 cdleme0cq 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( Q .\/ 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 6 oveq2i 
 |-  ( Q .\/ U ) = ( Q .\/ ( ( P .\/ Q ) ./\ W ) )
8 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> K e. HL )
9 simprrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> Q e. A )
10 hllat 
 |-  ( K e. HL -> K e. Lat )
11 10 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> K e. Lat )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
14 13 ad2antrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> P e. ( Base ` K ) )
15 12 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
16 9 15 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> Q e. ( Base ` K ) )
17 12 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
18 11 14 16 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
19 12 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
20 19 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> W e. ( Base ` K ) )
21 12 1 2 latlej2 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> Q .<_ ( P .\/ Q ) )
22 11 14 16 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> Q .<_ ( P .\/ Q ) )
23 12 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ Q .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( Q .\/ W ) ) )
24 8 9 18 20 22 23 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( Q .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( Q .\/ W ) ) )
25 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
26 1 2 25 4 5 lhpjat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q .\/ W ) = ( 1. ` K ) )
27 26 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( Q .\/ W ) = ( 1. ` K ) )
28 27 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ W ) ) = ( ( P .\/ Q ) ./\ ( 1. ` K ) ) )
29 hlol 
 |-  ( K e. HL -> K e. OL )
30 29 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> K e. OL )
31 12 3 25 olm11 
 |-  ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
32 30 18 31 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
33 24 28 32 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( Q .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) )
34 7 33 syl5eq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( Q .\/ U ) = ( P .\/ Q ) )