Metamath Proof Explorer

Theorem cdleme2

Description: Part of proof of Lemma E in Crawley p. 113. F represents f(r). W is the fiducial co-atom (hyperplane) w. Here we show that (r \/ f(r)) /\ w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012)

Ref Expression
Hypotheses cdleme1.l 
|- .<_ = ( le ` K )
cdleme1.j 
|- .\/ = ( join ` K )
cdleme1.m 
|- ./\ = ( meet ` K )
cdleme1.a 
|- A = ( Atoms ` K )
cdleme1.h 
|- H = ( LHyp ` K )
cdleme1.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdleme2 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ F ) ./\ W ) = U )

Proof

Step Hyp Ref Expression
1 cdleme1.l 
 |-  .<_ = ( le ` K )
2 cdleme1.j 
 |-  .\/ = ( join ` K )
3 cdleme1.m 
 |-  ./\ = ( meet ` K )
4 cdleme1.a 
 |-  A = ( Atoms ` K )
5 cdleme1.h 
 |-  H = ( LHyp ` K )
6 cdleme1.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme1.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 1 2 3 4 5 6 7 cdleme1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
9 8 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ F ) ./\ W ) = ( ( R .\/ U ) ./\ W ) )
10 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
11 simpr3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
12 hllat 
 |-  ( K e. HL -> K e. Lat )
13 12 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. Lat )
14 simpr1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
17 14 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. ( Base ` K ) )
18 simpr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. A )
19 15 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. ( Base ` K ) )
21 15 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
22 13 17 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
23 15 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
24 23 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> W e. ( Base ` K ) )
25 15 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
26 13 22 24 25 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
27 6 26 eqeltrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. ( Base ` K ) )
28 15 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
29 13 22 24 28 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
30 6 29 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> U .<_ W )
31 15 1 2 3 4 atmod4i2 
 |-  ( ( K e. HL /\ ( R e. A /\ U e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ U .<_ W ) -> ( ( R ./\ W ) .\/ U ) = ( ( R .\/ U ) ./\ W ) )
32 10 11 27 24 30 31 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R ./\ W ) .\/ U ) = ( ( R .\/ U ) ./\ W ) )
33 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
34 1 3 33 4 5 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R ./\ W ) = ( 0. ` K ) )
35 34 3ad2antr3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R ./\ W ) = ( 0. ` K ) )
36 35 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R ./\ W ) .\/ U ) = ( ( 0. ` K ) .\/ U ) )
37 hlol 
 |-  ( K e. HL -> K e. OL )
38 37 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. OL )
39 15 2 33 olj02 
 |-  ( ( K e. OL /\ U e. ( Base ` K ) ) -> ( ( 0. ` K ) .\/ U ) = U )
40 38 27 39 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( 0. ` K ) .\/ U ) = U )
41 36 40 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R ./\ W ) .\/ U ) = U )
42 9 32 41 3eqtr2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ F ) ./\ W ) = U )