This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdlemk28-3

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 14-Jul-2013)

Ref Expression
Hypotheses cdlemk3.b 
|- B = ( Base ` K )
cdlemk3.l 
|- .<_ = ( le ` K )
cdlemk3.j 
|- .\/ = ( join ` K )
cdlemk3.m 
|- ./\ = ( meet ` K )
cdlemk3.a 
|- A = ( Atoms ` K )
cdlemk3.h 
|- H = ( LHyp ` K )
cdlemk3.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk3.r 
|- R = ( ( trL ` K ) ` W )
cdlemk3.s 
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk3.u1 
|- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
Assertion cdlemk28-3 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> E. z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk3.b 
 |-  B = ( Base ` K )
2 cdlemk3.l 
 |-  .<_ = ( le ` K )
3 cdlemk3.j 
 |-  .\/ = ( join ` K )
4 cdlemk3.m 
 |-  ./\ = ( meet ` K )
5 cdlemk3.a 
 |-  A = ( Atoms ` K )
6 cdlemk3.h 
 |-  H = ( LHyp ` K )
7 cdlemk3.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk3.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk3.s 
 |-  S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10 cdlemk3.u1 
 |-  Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
11 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> F e. T )
13 simp21r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> F =/= ( _I |` B ) )
14 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> N e. T )
15 12 13 14 3jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) )
16 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G e. T )
17 simp22r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G =/= ( _I |` B ) )
18 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` F ) = ( R ` N ) )
19 16 17 18 3jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( G e. T /\ G =/= ( _I |` B ) /\ ( R ` F ) = ( R ` N ) ) )
20 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P e. A /\ -. P .<_ W ) )
21 1 2 3 4 5 6 7 8 9 10 cdlemk26b-3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> E. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( b Y G ) e. T ) )
22 11 15 19 20 21 syl31anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> E. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( b Y G ) e. T ) )
23 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
24 12 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> F e. T )
25 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> b e. T )
26 simp123 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> N e. T )
27 24 25 26 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ b e. T /\ N e. T ) )
28 16 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> G e. T )
29 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> a e. T )
30 28 29 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( G e. T /\ a e. T ) )
31 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
32 simp13r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` F ) = ( R ` N ) )
33 13 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> F =/= ( _I |` B ) )
34 simp3l1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> b =/= ( _I |` B ) )
35 32 33 34 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) )
36 17 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> G =/= ( _I |` B ) )
37 simp3r1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> a =/= ( _I |` B ) )
38 36 37 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( G =/= ( _I |` B ) /\ a =/= ( _I |` B ) ) )
39 simp3r3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` a ) =/= ( R ` G ) )
40 39 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` G ) =/= ( R ` a ) )
41 simp3r2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` a ) =/= ( R ` F ) )
42 simp3l2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` b ) =/= ( R ` F ) )
43 40 41 42 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( ( R ` G ) =/= ( R ` a ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` F ) ) )
44 simp3l3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` b ) =/= ( R ` G ) )
45 44 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( R ` G ) =/= ( R ` b ) )
46 1 2 3 4 5 6 7 8 9 10 cdlemk27-3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ b e. T /\ N e. T ) /\ ( G e. T /\ a e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ b =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ a =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` a ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` b ) ) ) -> ( b Y G ) = ( a Y G ) )
47 23 27 30 31 35 38 43 45 46 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( b e. T /\ a e. T ) /\ ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) ) -> ( b Y G ) = ( a Y G ) )
48 47 3exp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( b e. T /\ a e. T ) -> ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) -> ( b Y G ) = ( a Y G ) ) ) )
49 48 ralrimivv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> A. b e. T A. a e. T ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) -> ( b Y G ) = ( a Y G ) ) )
50 neeq1 
 |-  ( b = a -> ( b =/= ( _I |` B ) <-> a =/= ( _I |` B ) ) )
51 fveq2 
 |-  ( b = a -> ( R ` b ) = ( R ` a ) )
52 51 neeq1d 
 |-  ( b = a -> ( ( R ` b ) =/= ( R ` F ) <-> ( R ` a ) =/= ( R ` F ) ) )
53 51 neeq1d 
 |-  ( b = a -> ( ( R ` b ) =/= ( R ` G ) <-> ( R ` a ) =/= ( R ` G ) ) )
54 50 52 53 3anbi123d 
 |-  ( b = a -> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) <-> ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) )
55 oveq1 
 |-  ( b = a -> ( b Y G ) = ( a Y G ) )
56 54 55 reusv3 
 |-  ( E. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( b Y G ) e. T ) -> ( A. b e. T A. a e. T ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) -> ( b Y G ) = ( a Y G ) ) <-> E. z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) ) )
57 56 biimpd 
 |-  ( E. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( b Y G ) e. T ) -> ( A. b e. T A. a e. T ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( a =/= ( _I |` B ) /\ ( R ` a ) =/= ( R ` F ) /\ ( R ` a ) =/= ( R ` G ) ) ) -> ( b Y G ) = ( a Y G ) ) -> E. z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) ) )
58 22 49 57 sylc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> E. z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( b Y G ) ) )