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Metamath Proof Explorer

Theorem cdlemk27-3

Description: Part of proof of Lemma K of Crawley p. 118. Eliminate the P from the conclusion of cdlemk25-3 . (Contributed by NM, 10-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 cdlemk27-3 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( D Y G ) = ( C 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 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp221 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` F ) = ( R ` N ) )
13 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> G e. T )
14 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( F e. T /\ D e. T /\ N e. T ) )
15 simp3l3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` D ) =/= ( R ` F ) )
16 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` G ) =/= ( R ` D ) )
17 16 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` D ) =/= ( R ` G ) )
18 15 17 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) )
19 simp222 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> F =/= ( _I |` B ) )
20 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> G =/= ( _I |` B ) )
21 simp223 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> D =/= ( _I |` B ) )
22 19 20 21 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) )
23 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( P e. A /\ -. P .<_ W ) )
24 1 2 3 4 5 6 7 8 9 10 cdlemkuel-3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( D Y G ) e. T )
25 11 12 13 14 18 22 23 24 syl313anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( D Y G ) e. T )
26 simp121 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> F e. T )
27 simp13r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> C e. T )
28 simp123 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> N e. T )
29 simp3l2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` C ) =/= ( R ` F ) )
30 simp3l1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` G ) =/= ( R ` C ) )
31 30 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( R ` C ) =/= ( R ` G ) )
32 29 31 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` G ) ) )
33 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> C =/= ( _I |` B ) )
34 19 20 33 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) )
35 1 2 3 4 5 6 7 8 9 10 cdlemkuel-3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ C e. T /\ N e. T ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( C Y G ) e. T )
36 11 12 13 26 27 28 32 34 23 35 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( C Y G ) e. T )
37 1 2 3 4 5 6 7 8 9 10 cdlemk26-3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
38 2 5 6 7 cdlemd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( D Y G ) e. T /\ ( C Y G ) e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) ) -> ( D Y G ) = ( C Y G ) )
39 11 25 36 23 37 38 syl311anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( D Y G ) = ( C Y G ) )