Theorem cdlemk28-3 30347

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

Hypotheses

Ref	Expression
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

Ref	Expression
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 ) ) )

Distinct variable groups:   e, d, f, i, ./\   .<_ , i   .\/ , d, e, f, i   A, i   j, d, e, f, i, F   G, d, e, j   i, H   i, K   f, N, i   P, d, e, f, i   R, d, e, f, i   T, d, e, f, i   W, d, e, f, i, b   ./\ , j   .<_ , j   .\/ , j   A, j   j, F   j, H   j, K   j, N   P, j   R, j   b, d, S, e, j   T, j   j, W   F, d, e   .<_ , e   f, G, i   .<_ , b   A, b   z, b, B   F, b, z   G, b, z   H, b   K, b   N, b   P, b   R, b, z   T, b, z   W, b, z   Y, b, z   z, d, e, f, i, j

Allowed substitution hints:   A( z, e, f, d)   B( e, f, i, j, d)   P( z)   S( z, f, i)   H( z, e, f, d)   .\/ ( z, b)   K( z, e, f, d)   .<_ ( z, f, d)   ./\ ( z, b)   N( z, e, d)   Y( e, f, i, j, d)

Proof of Theorem cdlemk28-3

Step	Hyp	Ref	Expression
1		simp1 960	. . 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 ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp21l 1077	. . . 4 |- ( ( ( 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 )
3		simp21r 1078	. . . 4 |- ( ( ( 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 ) )
4		simp23 995	. . . 4 |- ( ( ( 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 )
5	2, 3, 4	3jca 1137	. . 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 ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) /\ N e. T ) )
6		simp22l 1079	. . . 4 |- ( ( ( 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 )
7		simp22r 1080	. . . 4 |- ( ( ( 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 ) )
8		simp3r 989	. . . 4 |- ( ( ( 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 ) )
9	6, 7, 8	3jca 1137	. . 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 ) ) ) -> ( G e. T /\ G =/= ( _I |` B ) /\ ( R ` F ) = ( R ` N ) ) )
10		simp3l 988	. . 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 ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11		cdlemk3.b	. . . 4 |- B = ( Base ` K )
12		cdlemk3.l	. . . 4 |- .<_ = ( le ` K )
13		cdlemk3.j	. . . 4 |- .\/ = ( join ` K )
14		cdlemk3.m	. . . 4 |- ./\ = ( meet ` K )
15		cdlemk3.a	. . . 4 |- A = ( Atoms ` K )
16		cdlemk3.h	. . . 4 |- H = ( LHyp ` K )
17		cdlemk3.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
18		cdlemk3.r	. . . 4 |- R = ( ( trL ` K ) ` W )
19		cdlemk3.s	. . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
20		cdlemk3.u1	. . . 4 |- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
21	11, 12, 13, 14, 15, 16, 17, 18, 19, 20	cdlemk26b-3 30344	. . 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	1, 5, 9, 10, 21	syl31anc 1190	. 2 |- ( ( ( 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 990	. . . . 5 |- ( ( ( ( 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	2	3ad2ant1 981	. . . . . 6 |- ( ( ( ( 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 986	. . . . . 6 |- ( ( ( ( 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 1094	. . . . . 6 |- ( ( ( ( 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 1137	. . . . 5 |- ( ( ( ( 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	6	3ad2ant1 981	. . . . . 6 |- ( ( ( ( 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 987	. . . . . 6 |- ( ( ( ( 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 520	. . . . 5 |- ( ( ( ( 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 1075	. . . . 5 |- ( ( ( ( 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 1076	. . . . . 6 |- ( ( ( ( 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	3	3ad2ant1 981	. . . . . 6 |- ( ( ( ( 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 1065	. . . . . 6 |- ( ( ( ( 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 1137	. . . . 5 |- ( ( ( ( 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	7	3ad2ant1 981	. . . . . 6 |- ( ( ( ( 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 1068	. . . . . 6 |- ( ( ( ( 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 520	. . . . 5 |- ( ( ( ( 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 1070	. . . . . . 7 |- ( ( ( ( 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 2504	. . . . . 6 |- ( ( ( ( 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 1069	. . . . . 6 |- ( ( ( ( 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 1066	. . . . . 6 |- ( ( ( ( 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 1137	. . . . 5 |- ( ( ( ( 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 1067	. . . . . 6 |- ( ( ( ( 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 2504	. . . . 5 |- ( ( ( ( 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	11, 12, 13, 14, 15, 16, 17, 18, 19, 20	cdlemk27-3 30346	. . . . 5 |- ( ( ( ( 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 1218	. . . 4 |- ( ( ( ( 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 1155	. . 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 ) ) ) -> ( ( 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 2609	. 2 |- ( ( ( 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 2429	. . . . 5 |- ( b = a -> ( b =/= ( _I |` B ) <-> a =/= ( _I |` B ) ) )
51		fveq2 5458	. . . . . 6 |- ( b = a -> ( R ` b ) = ( R ` a ) )
52	51	neeq1d 2434	. . . . 5 |- ( b = a -> ( ( R ` b ) =/= ( R ` F ) <-> ( R ` a ) =/= ( R ` F ) ) )
53	51	neeq1d 2434	. . . . 5 |- ( b = a -> ( ( R ` b ) =/= ( R ` G ) <-> ( R ` a ) =/= ( R ` G ) ) )
54	50, 52, 53	3anbi123d 1257	. . . 4 |- ( 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 5799	. . . 4 |- ( b = a -> ( b Y G ) = ( a Y G ) )
56	54, 55	reusv3 4514	. . 3 |- ( 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 200	. 2 |- ( 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 58	1 |- ( ( ( 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 ) ) )

Syntax hints:   -. wn 5   -> wi 6   /\ wa 360   /\ w3a 939   = wceq 1619   e. wcel 1621   =/= wne 2421   A.wral 2518   E.wrex 2519   class class class wbr 3997   e. cmpt 4051   _I cid 4276   `'ccnv 4660   |` cres 4663   o. ccom 4665   ` cfv 4673  (class class class)co 5792   e. cmpt2 5794   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Atomscatm 28703   HLchlt 28790   LHypclh 29423   LTrncltrn 29540   trLctrl 29597

This theorem is referenced by:  cdlemk29-3  30350

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598