Theorem cdlemk28-3 36196

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

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . 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 1178	. . . 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 1179	. . . 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 1096	. . . 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 1242	. . 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 1180	. . . 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 1181	. . . 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 1090	. . . 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 1242	. . 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 1089	. . 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 36193	. . 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 1329	. 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 1091	. . . . 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 1082	. . . . . 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 1087	. . . . . 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 1195	. . . . . 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 1242	. . . . 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 1082	. . . . . 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 1088	. . . . . 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 554	. . . . 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 1176	. . . . 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 1177	. . . . . 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 1082	. . . . . 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 1166	. . . . . 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 1242	. . . . 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 1082	. . . . . 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 1169	. . . . . 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 554	. . . . 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 1171	. . . . . . 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 2849	. . . . . 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 1170	. . . . . 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 1167	. . . . . 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 1242	. . . . 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 1168	. . . . . 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 2849	. . . . 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 36195	. . . . 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 1357	. . . 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 1264	. . 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 2970	. 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 2856	. . . . 5 |- ( b = a -> ( b =/= ( _I |` B ) <-> a =/= ( _I |` B ) ) )
51		fveq2 6191	. . . . . 6 |- ( b = a -> ( R ` b ) = ( R ` a ) )
52	51	neeq1d 2853	. . . . 5 |- ( b = a -> ( ( R ` b ) =/= ( R ` F ) <-> ( R ` a ) =/= ( R ` F ) ) )
53	51	neeq1d 2853	. . . . 5 |- ( b = a -> ( ( R ` b ) =/= ( R ` G ) <-> ( R ` a ) =/= ( R ` G ) ) )
54	50, 52, 53	3anbi123d 1399	. . . 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 6657	. . . 4 |- ( b = a -> ( b Y G ) = ( a Y G ) )
56	54, 55	reusv3 4876	. . 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 219	. 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 65	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 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemk29-3  36199