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 ) ) )

Metamath Proof Explorer

Theorem cdlemk28-3

Proof