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Metamath Proof Explorer

Theorem cdleme20c

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, second line. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 15-Nov-2012)

Ref Expression
Hypotheses cdleme19.l 
|- .<_ = ( le ` K )
cdleme19.j 
|- .\/ = ( join ` K )
cdleme19.m 
|- ./\ = ( meet ` K )
cdleme19.a 
|- A = ( Atoms ` K )
cdleme19.h 
|- H = ( LHyp ` K )
cdleme19.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme19.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme20.v 
|- V = ( ( S .\/ T ) ./\ W )
Assertion cdleme20c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( D .\/ Y ) = ( ( ( R .\/ S ) .\/ T ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 cdleme19.l 
 |-  .<_ = ( le ` K )
2 cdleme19.j 
 |-  .\/ = ( join ` K )
3 cdleme19.m 
 |-  ./\ = ( meet ` K )
4 cdleme19.a 
 |-  A = ( Atoms ` K )
5 cdleme19.h 
 |-  H = ( LHyp ` K )
6 cdleme19.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme19.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme19.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme19.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme19.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme20.v 
 |-  V = ( ( S .\/ T ) ./\ W )
12 9 10 oveq12i 
 |-  ( D .\/ Y ) = ( ( ( R .\/ S ) ./\ W ) .\/ ( ( R .\/ T ) ./\ W ) )
13 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
14 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
15 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
18 13 14 15 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
19 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
20 16 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
21 19 20 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
22 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> R .<_ ( R .\/ S ) )
23 13 14 15 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( R .\/ S ) )
24 16 1 2 3 4 atmod2i1 
 |-  ( ( K e. HL /\ ( R e. A /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ R .<_ ( R .\/ S ) ) -> ( ( ( R .\/ S ) ./\ W ) .\/ R ) = ( ( R .\/ S ) ./\ ( W .\/ R ) ) )
25 13 14 18 21 23 24 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) ./\ W ) .\/ R ) = ( ( R .\/ S ) ./\ ( W .\/ R ) ) )
26 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
27 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
28 1 2 27 4 5 lhpjat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( W .\/ R ) = ( 1. ` K ) )
29 13 19 26 28 syl21anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( W .\/ R ) = ( 1. ` K ) )
30 29 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ ( W .\/ R ) ) = ( ( R .\/ S ) ./\ ( 1. ` K ) ) )
31 hlol 
 |-  ( K e. HL -> K e. OL )
32 13 31 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. OL )
33 16 3 27 olm11 
 |-  ( ( K e. OL /\ ( R .\/ S ) e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ ( 1. ` K ) ) = ( R .\/ S ) )
34 32 18 33 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ ( 1. ` K ) ) = ( R .\/ S ) )
35 25 30 34 3eqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) = ( ( ( R .\/ S ) ./\ W ) .\/ R ) )
36 35 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) .\/ T ) = ( ( ( ( R .\/ S ) ./\ W ) .\/ R ) .\/ T ) )
37 simp22r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ W )
38 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
39 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
40 eqid 
 |-  ( ( R .\/ S ) ./\ W ) = ( ( R .\/ S ) ./\ W )
41 1 2 3 4 5 40 cdlemeda 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. A )
42 13 19 15 37 14 38 39 41 syl223anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) e. A )
43 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> T e. A )
44 2 4 hlatjass 
 |-  ( ( K e. HL /\ ( ( ( R .\/ S ) ./\ W ) e. A /\ R e. A /\ T e. A ) ) -> ( ( ( ( R .\/ S ) ./\ W ) .\/ R ) .\/ T ) = ( ( ( R .\/ S ) ./\ W ) .\/ ( R .\/ T ) ) )
45 13 42 14 43 44 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( ( R .\/ S ) ./\ W ) .\/ R ) .\/ T ) = ( ( ( R .\/ S ) ./\ W ) .\/ ( R .\/ T ) ) )
46 36 45 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) .\/ T ) = ( ( ( R .\/ S ) ./\ W ) .\/ ( R .\/ T ) ) )
47 46 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) .\/ T ) ./\ W ) = ( ( ( ( R .\/ S ) ./\ W ) .\/ ( R .\/ T ) ) ./\ W ) )
48 16 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ T e. A ) -> ( R .\/ T ) e. ( Base ` K ) )
49 13 14 43 48 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ T ) e. ( Base ` K ) )
50 13 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
51 16 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
52 50 18 21 51 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) ./\ W ) .<_ W )
53 16 1 2 3 4 atmod1i1 
 |-  ( ( K e. HL /\ ( ( ( R .\/ S ) ./\ W ) e. A /\ ( R .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ ( ( R .\/ S ) ./\ W ) .<_ W ) -> ( ( ( R .\/ S ) ./\ W ) .\/ ( ( R .\/ T ) ./\ W ) ) = ( ( ( ( R .\/ S ) ./\ W ) .\/ ( R .\/ T ) ) ./\ W ) )
54 13 42 49 21 52 53 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) ./\ W ) .\/ ( ( R .\/ T ) ./\ W ) ) = ( ( ( ( R .\/ S ) ./\ W ) .\/ ( R .\/ T ) ) ./\ W ) )
55 47 54 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) .\/ T ) ./\ W ) = ( ( ( R .\/ S ) ./\ W ) .\/ ( ( R .\/ T ) ./\ W ) ) )
56 12 55 eqtr4id 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ T e. A ) /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( D .\/ Y ) = ( ( ( R .\/ S ) .\/ T ) ./\ W ) )