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Metamath Proof Explorer

Theorem cdlemk11

Description: Part of proof of Lemma K of Crawley p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-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 cdlemk11 
|- ( ( ( ( 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 ) .\/ ( R ` ( X o. `' G ) ) ) )

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 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 )
12 11 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 )
13 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 ) )
14 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 )
15 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 ) )
16 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 ) )
17 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 ) )
18 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 ) )
19 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 ) )
20 1 2 3 4 5 6 7 8 9 cdlemksat 
 |-  ( ( ( ( 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 ) e. A )
21 13 14 15 16 17 18 19 20 syl133anc 
 |-  ( ( ( ( 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 ) e. A )
22 1 4 atbase 
 |-  ( ( ( S ` G ) ` P ) e. A -> ( ( S ` G ) ` P ) e. B )
23 21 22 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 ` G ) ` P ) e. B )
24 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 ) )
25 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 )
26 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 )
27 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 ) )
28 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 ) )
29 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 )
30 24 25 26 14 15 16 17 27 28 29 syl333anc 
 |-  ( ( ( ( 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 )
31 1 4 atbase 
 |-  ( ( ( S ` X ) ` P ) e. A -> ( ( S ` X ) ` P ) e. B )
32 30 31 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 )
33 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 )
34 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 )
35 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 )
36 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 )
37 11 33 25 34 26 35 36 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 )
38 1 3 latjcl 
 |-  ( ( K e. Lat /\ ( ( S ` X ) ` P ) e. B /\ V e. B ) -> ( ( ( S ` X ) ` P ) .\/ V ) e. B )
39 12 32 37 38 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 ) e. B )
40 5 6 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> `' G e. T )
41 24 34 40 syl2anc 
 |-  ( ( ( ( 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 )
42 5 6 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ `' G e. T ) -> ( X o. `' G ) e. T )
43 24 26 41 42 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 o. `' G ) e. T )
44 1 5 6 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X o. `' G ) e. T ) -> ( R ` ( X o. `' G ) ) e. B )
45 24 43 44 syl2anc 
 |-  ( ( ( ( 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. `' G ) ) e. B )
46 1 3 latjcl 
 |-  ( ( K e. Lat /\ ( ( S ` X ) ` P ) e. B /\ ( R ` ( X o. `' G ) ) e. B ) -> ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) e. B )
47 12 32 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 ) .\/ ( R ` ( X o. `' G ) ) ) e. B )
48 1 2 3 4 5 6 7 8 9 10 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 ) )
49 1 2 3 4 5 6 7 8 10 cdlemk10 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> V .<_ ( R ` ( X o. `' G ) ) )
50 11 33 25 34 26 15 49 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 .<_ ( R ` ( X o. `' G ) ) )
51 1 2 3 latjlej2 
 |-  ( ( K e. Lat /\ ( V e. B /\ ( R ` ( X o. `' G ) ) e. B /\ ( ( S ` X ) ` P ) e. B ) ) -> ( V .<_ ( R ` ( X o. `' G ) ) -> ( ( ( S ` X ) ` P ) .\/ V ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )
52 12 37 45 32 51 syl13anc 
 |-  ( ( ( ( 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 .<_ ( R ` ( X o. `' G ) ) -> ( ( ( S ` X ) ` P ) .\/ V ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )
53 50 52 mpd 
 |-  ( ( ( ( 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 ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
54 1 2 12 23 39 47 48 53 lattrd 
 |-  ( ( ( ( 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 ) .\/ ( R ` ( X o. `' G ) ) ) )