Metamath Proof Explorer

Theorem cdleme17c

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. C represents s_1. We show, in their notation, (p \/ q) /\ (q \/ s_1)=q. (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses cdleme17.l 
|- .<_ = ( le ` K )
cdleme17.j 
|- .\/ = ( join ` K )
cdleme17.m 
|- ./\ = ( meet ` K )
cdleme17.a 
|- A = ( Atoms ` K )
cdleme17.h 
|- H = ( LHyp ` K )
cdleme17.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme17.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme17.g 
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme17.c 
|- C = ( ( P .\/ S ) ./\ W )
Assertion cdleme17c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) = Q )

Proof

Step Hyp Ref Expression
1 cdleme17.l 
 |-  .<_ = ( le ` K )
2 cdleme17.j 
 |-  .\/ = ( join ` K )
3 cdleme17.m 
 |-  ./\ = ( meet ` K )
4 cdleme17.a 
 |-  A = ( Atoms ` K )
5 cdleme17.h 
 |-  H = ( LHyp ` K )
6 cdleme17.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme17.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme17.g 
 |-  G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
9 cdleme17.c 
 |-  C = ( ( P .\/ S ) ./\ W )
10 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
11 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
12 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
13 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
14 10 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
15 14 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) = ( ( Q .\/ P ) ./\ ( Q .\/ C ) ) )
16 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
17 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. P .<_ W )
18 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
19 10 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
20 eqid 
 |-  ( Base ` K ) = ( Base ` K )
21 20 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
22 18 21 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. ( Base ` K ) )
23 20 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
24 11 23 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) )
25 20 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
26 12 25 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. ( Base ` K ) )
27 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
28 20 1 2 latnlej1l 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P )
29 28 necomd 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S )
30 19 22 24 26 27 29 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= S )
31 1 2 3 4 5 9 cdleme9a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> C e. A )
32 10 16 11 17 18 30 31 syl222anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> C e. A )
33 1 2 3 4 5 6 7 8 9 cdleme17b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. C .<_ ( P .\/ Q ) )
34 1 2 3 4 2llnma1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ C e. A ) /\ -. C .<_ ( P .\/ Q ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ C ) ) = Q )
35 10 11 12 32 33 34 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ C ) ) = Q )
36 15 35 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) = Q )