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Metamath Proof Explorer

Theorem cdleme20d

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, 17-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 cdleme20d 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( F .\/ G ) ./\ ( D .\/ Y ) ) = V )

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 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
13 hlol 
 |-  ( K e. HL -> K e. OL )
14 12 13 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. OL )
15 12 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
16 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
17 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> P e. A )
18 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> Q e. A )
19 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
20 eqid 
 |-  ( Base ` K ) = ( Base ` K )
21 1 2 3 4 5 6 7 20 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
22 12 16 17 18 19 21 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> F e. ( Base ` K ) )
23 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> T e. A )
24 1 2 3 4 5 6 8 20 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ T e. A ) ) -> G e. ( Base ` K ) )
25 12 16 17 18 23 24 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> G e. ( Base ` K ) )
26 20 2 latjcl 
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ G e. ( Base ` K ) ) -> ( F .\/ G ) e. ( Base ` K ) )
27 15 22 25 26 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ G ) e. ( Base ` K ) )
28 20 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
29 16 28 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. ( Base ` K ) )
30 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
31 20 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
32 12 30 19 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
33 20 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
34 23 33 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> T e. ( Base ` K ) )
35 20 2 latjcl 
 |-  ( ( K e. Lat /\ ( R .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( R .\/ S ) .\/ T ) e. ( Base ` K ) )
36 15 32 34 35 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ S ) .\/ T ) e. ( Base ` K ) )
37 20 3 latmassOLD 
 |-  ( ( K e. OL /\ ( ( F .\/ G ) e. ( Base ` K ) /\ W e. ( Base ` K ) /\ ( ( R .\/ S ) .\/ T ) e. ( Base ` K ) ) ) -> ( ( ( F .\/ G ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) = ( ( F .\/ G ) ./\ ( W ./\ ( ( R .\/ S ) .\/ T ) ) ) )
38 14 27 29 36 37 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( F .\/ G ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) = ( ( F .\/ G ) ./\ ( W ./\ ( ( R .\/ S ) .\/ T ) ) ) )
39 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> S .<_ ( R .\/ S ) )
40 12 30 19 39 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> S .<_ ( R .\/ S ) )
41 20 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
42 19 41 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. ( Base ` K ) )
43 20 1 2 latjlej1 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) ) -> ( S .<_ ( R .\/ S ) -> ( S .\/ T ) .<_ ( ( R .\/ S ) .\/ T ) ) )
44 15 42 32 34 43 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S .<_ ( R .\/ S ) -> ( S .\/ T ) .<_ ( ( R .\/ S ) .\/ T ) ) )
45 40 44 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S .\/ T ) .<_ ( ( R .\/ S ) .\/ T ) )
46 20 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
47 12 19 23 46 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
48 20 1 3 latleeqm1 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ ( ( R .\/ S ) .\/ T ) e. ( Base ` K ) ) -> ( ( S .\/ T ) .<_ ( ( R .\/ S ) .\/ T ) <-> ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) = ( S .\/ T ) ) )
49 15 47 36 48 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ T ) .<_ ( ( R .\/ S ) .\/ T ) <-> ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) = ( S .\/ T ) ) )
50 45 49 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) = ( S .\/ T ) )
51 50 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) ./\ W ) = ( ( S .\/ T ) ./\ W ) )
52 11 51 eqtr4id 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> V = ( ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) ./\ W ) )
53 20 3 latm32 
 |-  ( ( K e. OL /\ ( ( S .\/ T ) e. ( Base ` K ) /\ ( ( R .\/ S ) .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) ./\ W ) = ( ( ( S .\/ T ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) )
54 14 47 36 29 53 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) ./\ W ) = ( ( ( S .\/ T ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) )
55 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
56 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
57 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( T e. A /\ -. T .<_ W ) )
58 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( P =/= Q /\ S =/= T ) )
59 simp32l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
60 simp32r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> -. T .<_ ( P .\/ Q ) )
61 1 2 3 4 5 6 7 8 cdleme16 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( F .\/ G ) ./\ W ) )
62 55 56 57 58 59 60 61 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( F .\/ G ) ./\ W ) )
63 62 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ T ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) = ( ( ( F .\/ G ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) )
64 54 63 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( S .\/ T ) ./\ ( ( R .\/ S ) .\/ T ) ) ./\ W ) = ( ( ( F .\/ G ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) )
65 52 64 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> V = ( ( ( F .\/ G ) ./\ W ) ./\ ( ( R .\/ S ) .\/ T ) ) )
66 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
67 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
68 1 2 3 4 5 6 7 8 9 10 11 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 ) )
69 12 16 66 56 23 59 67 68 syl232anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( D .\/ Y ) = ( ( ( R .\/ S ) .\/ T ) ./\ W ) )
70 20 3 latmcom 
 |-  ( ( K e. Lat /\ ( ( R .\/ S ) .\/ T ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( R .\/ S ) .\/ T ) ./\ W ) = ( W ./\ ( ( R .\/ S ) .\/ T ) ) )
71 15 36 29 70 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( ( R .\/ S ) .\/ T ) ./\ W ) = ( W ./\ ( ( R .\/ S ) .\/ T ) ) )
72 69 71 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( D .\/ Y ) = ( W ./\ ( ( R .\/ S ) .\/ T ) ) )
73 72 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( F .\/ G ) ./\ ( D .\/ Y ) ) = ( ( F .\/ G ) ./\ ( W ./\ ( ( R .\/ S ) .\/ T ) ) ) )
74 38 65 73 3eqtr4rd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( F .\/ G ) ./\ ( D .\/ Y ) ) = V )