Metamath Proof Explorer

Theorem cdlemk7

Description: Part of proof of Lemma K of Crawley p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013)

Ref Expression
Hypotheses cdlemk.b 
|- B = ( Base ` K )
cdlemk.l 
|- .<_ = ( le ` K )
cdlemk.j 
|- .\/ = ( join ` K )
cdlemk.a 
|- A = ( Atoms ` K )
cdlemk.h 
|- H = ( LHyp ` K )
cdlemk.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk.r 
|- R = ( ( trL ` K ) ` W )
cdlemk.m 
|- ./\ = ( meet ` K )
cdlemk.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk.v 
|- V = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) )
Assertion cdlemk7 
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) .<_ ( ( ( S ` X ) ` P ) .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdlemk.b 
 |-  B = ( Base ` K )
2 cdlemk.l 
 |-  .<_ = ( le ` K )
3 cdlemk.j 
 |-  .\/ = ( join ` K )
4 cdlemk.a 
 |-  A = ( Atoms ` K )
5 cdlemk.h 
 |-  H = ( LHyp ` K )
6 cdlemk.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemk.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemk.m 
 |-  ./\ = ( meet ` K )
9 cdlemk.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk.v 
 |-  V = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) )
11 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) )
12 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
13 simp311 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> F =/= ( _I |` B ) )
14 simp312 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> G =/= ( _I |` B ) )
15 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( R ` G ) =/= ( R ` F ) )
16 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( R ` X ) =/= ( R ` F ) )
17 15 16 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) )
18 1 2 3 4 5 6 7 8 cdlemk6 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) )
19 11 12 13 14 17 18 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) )
20 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> N e. T )
21 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( P e. A /\ -. P .<_ W ) )
22 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( R ` F ) = ( R ` N ) )
23 20 21 22 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
24 1 2 3 4 5 6 7 8 9 cdlemksv2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
25 11 23 13 14 15 24 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
26 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
27 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> G e. T )
28 2 3 4 5 6 7 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
29 26 27 21 28 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
30 29 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
31 25 30 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
32 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> K e. HL )
33 32 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> K e. Lat )
34 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> F e. T )
35 simp21r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> X e. T )
36 26 34 35 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ X e. T ) )
37 simp313 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> X =/= ( _I |` B ) )
38 1 2 3 4 5 6 7 8 9 cdlemksat 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ X e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) e. A )
39 36 23 13 37 16 38 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) e. A )
40 1 4 atbase 
 |-  ( ( ( S ` X ) ` P ) e. A -> ( ( S ` X ) ` P ) e. B )
41 39 40 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) e. B )
42 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> W e. H )
43 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> P e. A )
44 1 2 3 4 5 6 7 8 10 cdlemkvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ P e. A ) -> V e. B )
45 32 42 34 27 35 43 44 syl231anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> V e. B )
46 1 3 latjcom 
 |-  ( ( K e. Lat /\ ( ( S ` X ) ` P ) e. B /\ V e. B ) -> ( ( ( S ` X ) ` P ) .\/ V ) = ( V .\/ ( ( S ` X ) ` P ) ) )
47 33 41 45 46 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( ( S ` X ) ` P ) .\/ V ) = ( V .\/ ( ( S ` X ) ` P ) ) )
48 10 a1i 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> V = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) )
49 1 2 3 4 5 6 7 8 9 cdlemksv2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ X e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) = ( ( P .\/ ( R ` X ) ) ./\ ( ( N ` P ) .\/ ( R ` ( X o. `' F ) ) ) ) )
50 36 23 13 37 16 49 syl113anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) = ( ( P .\/ ( R ` X ) ) ./\ ( ( N ` P ) .\/ ( R ` ( X o. `' F ) ) ) ) )
51 2 3 4 5 6 7 trljat1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` X ) ) = ( P .\/ ( X ` P ) ) )
52 26 35 21 51 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` X ) ) = ( P .\/ ( X ` P ) ) )
53 2 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A )
54 26 35 43 53 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( X ` P ) e. A )
55 3 4 hlatjcom 
 |-  ( ( K e. HL /\ ( X ` P ) e. A /\ P e. A ) -> ( ( X ` P ) .\/ P ) = ( P .\/ ( X ` P ) ) )
56 32 54 43 55 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( X ` P ) .\/ P ) = ( P .\/ ( X ` P ) ) )
57 52 56 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` X ) ) = ( ( X ` P ) .\/ P ) )
58 2 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
59 26 20 43 58 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( N ` P ) e. A )
60 35 34 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( X e. T /\ F e. T ) )
61 4 5 6 7 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ F e. T ) /\ ( R ` X ) =/= ( R ` F ) ) -> ( R ` ( X o. `' F ) ) e. A )
62 26 60 16 61 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( R ` ( X o. `' F ) ) e. A )
63 3 4 hlatjcom 
 |-  ( ( K e. HL /\ ( N ` P ) e. A /\ ( R ` ( X o. `' F ) ) e. A ) -> ( ( N ` P ) .\/ ( R ` ( X o. `' F ) ) ) = ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) )
64 32 59 62 63 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( N ` P ) .\/ ( R ` ( X o. `' F ) ) ) = ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) )
65 57 64 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( P .\/ ( R ` X ) ) ./\ ( ( N ` P ) .\/ ( R ` ( X o. `' F ) ) ) ) = ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) )
66 50 65 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) = ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) )
67 48 66 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( V .\/ ( ( S ` X ) ` P ) ) = ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) )
68 47 67 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( ( S ` X ) ` P ) .\/ V ) = ( ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) .\/ ( ( ( X ` P ) .\/ P ) ./\ ( ( R ` ( X o. `' F ) ) .\/ ( N ` P ) ) ) ) )
69 19 31 68 3brtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) .<_ ( ( ( S ` X ) ` P ) .\/ V ) )