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 ) )

Metamath Proof Explorer

Theorem cdlemk27-3

Proof