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Metamath Proof Explorer

Theorem cdleme22g

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 6th and 7th lines on p. 115. F , G represent f(s), f(t) respectively. If s <_ t \/ v and -. s <_ p \/ q, then f(s) <_ f(t) \/ v. (Contributed by NM, 6-Dec-2012)

Ref Expression
Hypotheses cdleme22.l 
|- .<_ = ( le ` K )
cdleme22.j 
|- .\/ = ( join ` K )
cdleme22.m 
|- ./\ = ( meet ` K )
cdleme22.a 
|- A = ( Atoms ` K )
cdleme22.h 
|- H = ( LHyp ` K )
cdleme22g.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme22g.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme22g.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
Assertion cdleme22g 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( G .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdleme22.l 
 |-  .<_ = ( le ` K )
2 cdleme22.j 
 |-  .\/ = ( join ` K )
3 cdleme22.m 
 |-  ./\ = ( meet ` K )
4 cdleme22.a 
 |-  A = ( Atoms ` K )
5 cdleme22.h 
 |-  H = ( LHyp ` K )
6 cdleme22g.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme22g.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme22g.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. HL )
10 9 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. Lat )
11 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
13 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
14 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S e. A /\ -. S .<_ W ) )
15 simp133 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> P =/= Q )
16 simp132 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. S .<_ ( P .\/ Q ) )
17 1 2 3 4 5 6 7 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A )
18 11 12 13 14 15 16 17 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F e. A )
19 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( T e. A /\ -. T .<_ W ) )
20 simp131 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. T .<_ ( P .\/ Q ) )
21 1 2 3 4 5 6 8 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ -. T .<_ ( P .\/ Q ) ) ) -> G e. A )
22 11 12 13 19 15 20 21 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> G e. A )
23 eqid 
 |-  ( Base ` K ) = ( Base ` K )
24 23 2 4 hlatjcl 
 |-  ( ( K e. HL /\ F e. A /\ G e. A ) -> ( F .\/ G ) e. ( Base ` K ) )
25 9 18 22 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( F .\/ G ) e. ( Base ` K ) )
26 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> W e. H )
27 23 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
28 26 27 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> W e. ( Base ` K ) )
29 23 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( F .\/ G ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( F .\/ G ) ./\ W ) .<_ ( F .\/ G ) )
30 10 25 28 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( ( F .\/ G ) ./\ W ) .<_ ( F .\/ G ) )
31 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( V e. A /\ V .<_ W ) )
32 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( S =/= T /\ S .<_ ( T .\/ V ) ) )
33 1 2 3 4 5 cdleme22d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( V e. A /\ V .<_ W ) ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) ) -> V = ( ( S .\/ T ) ./\ W ) )
34 11 14 19 31 32 33 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V = ( ( S .\/ T ) ./\ W ) )
35 simp32l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> S =/= T )
36 15 35 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( P =/= Q /\ S =/= T ) )
37 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 ) )
38 11 12 13 14 19 36 16 20 37 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( ( S .\/ T ) ./\ W ) = ( ( F .\/ G ) ./\ W ) )
39 34 38 eqtr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( ( F .\/ G ) ./\ W ) = V )
40 2 4 hlatjcom 
 |-  ( ( K e. HL /\ F e. A /\ G e. A ) -> ( F .\/ G ) = ( G .\/ F ) )
41 9 18 22 40 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( F .\/ G ) = ( G .\/ F ) )
42 30 39 41 3brtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V .<_ ( G .\/ F ) )
43 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
44 9 43 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> K e. CvLat )
45 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V e. A )
46 simp33r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V .<_ W )
47 1 2 3 4 5 6 8 cdleme3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ -. T .<_ ( P .\/ Q ) ) ) -> -. G .<_ W )
48 11 12 13 19 15 20 47 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> -. G .<_ W )
49 nbrne2 
 |-  ( ( V .<_ W /\ -. G .<_ W ) -> V =/= G )
50 46 48 49 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> V =/= G )
51 1 2 4 cvlatexch1 
 |-  ( ( K e. CvLat /\ ( V e. A /\ F e. A /\ G e. A ) /\ V =/= G ) -> ( V .<_ ( G .\/ F ) -> F .<_ ( G .\/ V ) ) )
52 44 45 18 22 50 51 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> ( V .<_ ( G .\/ F ) -> F .<_ ( G .\/ V ) ) )
53 42 52 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( T e. A /\ -. T .<_ W ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ P =/= Q ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S =/= T /\ S .<_ ( T .\/ V ) ) /\ ( V e. A /\ V .<_ W ) ) ) -> F .<_ ( G .\/ V ) )