Theorem cdlemk22 37035

Description: Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 5-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk2.b	|- B = ( Base ` K )
cdlemk2.l	|- .<_ = ( le ` K )
cdlemk2.j	|- .\/ = ( join ` K )
cdlemk2.m	|- ./\ = ( meet ` K )
cdlemk2.a	|- A = ( Atoms ` K )
cdlemk2.h	|- H = ( LHyp ` K )
cdlemk2.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk2.r	|- R = ( ( trL ` K ) ` W )
cdlemk2.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk2.q	|- Q = ( S ` C )
cdlemk2.v	|- V = ( d e. T |-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
cdlemk2a.o	|- O = ( S ` D )
cdlemk2.u	|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )

Assertion

Ref

Expression

cdlemk22

|- ( ( ( ( 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 ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( V ` G ) ` P ) )

Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   C, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i   ./\ , d   .\/ , d   C, d, k   G, d, k   Q, d   P, d   R, d   T, d   W, d   ./\ , k   .<_ , k   .\/ , k   A, k   C, k   k, F   k, H   k, K   k, N   Q, k   P, k   R, k   T, k   k, W   F, d   i, G, f, k   D, k   i, d, D, f   e, j, ./\   .<_ , e, j   .\/ , e, j   A, j   C, e, j   D, e, j   e, F, j   e, G, j   j, H   j, K   j, N   e, O, j   P, e, j   R, e, j   T, e, j   e, W, j, f, i

Allowed substitution hints:   A( e, f, d)   B( e, f, i, j, k, d)   Q( e, f, i, j)   S( e, f, i, j, k, d)   U( e, f, i, j, k, d)   H( e, f, d)   K( e, f, d)   .<_ ( f, d)   N( e, d)   O( f, i, k, d)   V( e, f, i, j, k, d)

Proof of Theorem cdlemk22

Step	Hyp	Ref	Expression
1		simp11 1024	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp212 1133	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> G e. T )
3		simp22 1028	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
4		cdlemk2.l	. . . . 5 |- .<_ = ( le ` K )
5		cdlemk2.j	. . . . 5 |- .\/ = ( join ` K )
6		cdlemk2.a	. . . . 5 |- A = ( Atoms ` K )
7		cdlemk2.h	. . . . 5 |- H = ( LHyp ` K )
8		cdlemk2.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
9		cdlemk2.r	. . . . 5 |- R = ( ( trL ` K ) ` W )
10	4, 5, 6, 7, 8, 9	trljat1 36307	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
11	1, 2, 3, 10	syl3anc 1226	. . 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 ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
12		simp1 994	. . . . . 6 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
13		simp211 1132	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> N e. T )
14		simp213 1134	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> C e. T )
15	13, 14	jca 530	. . . . . 6 |- ( ( ( ( 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 ` D ) ) ) ) -> ( N e. T /\ C e. T ) )
16		simp23 1029	. . . . . 6 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` F ) = ( R ` N ) )
17		simp311 1141	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> F =/= ( _I |` B ) )
18		simp312 1142	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> D =/= ( _I |` B ) )
19		simp321 1144	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> C =/= ( _I |` B ) )
20	17, 18, 19	3jca 1174	. . . . . 6 |- ( ( ( ( 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 ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) )
21		simp331 1147	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
22		simp323 1146	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` C ) =/= ( R ` F ) )
23		simp333 1149	. . . . . . 7 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` C ) =/= ( R ` D ) )
24	21, 22, 23	3jca 1174	. . . . . 6 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) )
25		cdlemk2.b	. . . . . . 7 |- B = ( Base ` K )
26		cdlemk2.m	. . . . . . 7 |- ./\ = ( meet ` K )
27		cdlemk2.s	. . . . . . 7 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
28		cdlemk2a.o	. . . . . . 7 |- O = ( S ` D )
29		cdlemk2.u	. . . . . . 7 |- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) )
30		cdlemk2.q	. . . . . . 7 |- Q = ( S ` C )
31	25, 4, 5, 26, 6, 7, 8, 9, 27, 28, 29, 30	cdlemk20 37016	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( U ` C ) ` P ) = ( Q ` P ) )
32	12, 15, 3, 16, 20, 24, 31	syl132anc 1244	. . . . 5 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( U ` C ) ` P ) = ( Q ` P ) )
33	32	eqcomd 2462	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( Q ` P ) = ( ( U ` C ) ` P ) )
34	7, 8, 9	trlcocnv 36862	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ C e. T ) -> ( R ` ( G o. `' C ) ) = ( R ` ( C o. `' G ) ) )
35	1, 2, 14, 34	syl3anc 1226	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` ( G o. `' C ) ) = ( R ` ( C o. `' G ) ) )
36	33, 35	oveq12d 6288	. . 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 ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( Q ` P ) .\/ ( R ` ( G o. `' C ) ) ) = ( ( ( U ` C ) ` P ) .\/ ( R ` ( C o. `' G ) ) ) )
37	11, 36	oveq12d 6288	. 2 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' C ) ) ) ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( U ` C ) ` P ) .\/ ( R ` ( C o. `' G ) ) ) ) )
38		simp12 1025	. . 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 ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> F e. T )
39		simp322 1145	. . . . 5 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` G ) =/= ( R ` C ) )
40	39	necomd 2725	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( R ` C ) =/= ( R ` G ) )
41	22, 40	jca 530	. . 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 ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` C ) =/= ( R ` G ) ) )
42		simp313 1143	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> G =/= ( _I |` B ) )
43	17, 42, 19	3jca 1174	. . 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 ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) )
44		cdlemk2.v	. . . 4 |- V = ( d e. T |-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) )
45	25, 4, 5, 26, 6, 7, 8, 9, 27, 30, 44	cdlemkuv2-2 37027	. . 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 ) ) ) -> ( ( V ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' C ) ) ) ) )
46	1, 16, 2, 38, 14, 13, 41, 43, 3, 45	syl333anc 1258	. 2 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( V ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( G o. `' C ) ) ) ) )
47		simp31 1030	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
48	19, 39	jca 530	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) ) )
49		simp33 1032	. . . 4 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) )
50	47, 48, 49	3jca 1174	. . 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 ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) ) )
51	25, 4, 5, 26, 6, 7, 8, 9, 27, 28, 29	cdlemk12u 37014	. . 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 ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( U ` C ) ` P ) .\/ ( R ` ( C o. `' G ) ) ) ) )
52	50, 51	syld3an3 1271	. 2 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( U ` C ) ` P ) .\/ ( R ` ( C o. `' G ) ) ) ) )
53	37, 46, 52	3eqtr4rd 2506	1 |- ( ( ( ( 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 ` D ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( V ` G ) ` P ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   class class class wbr 4439   |-> cmpt 4497   _I cid 4779   `'ccnv 4987   |` cres 4990   o. ccom 4992   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-riotaBAD 35100

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-undef 6994  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lvols 35640  df-lines 35641  df-psubsp 35643  df-pmap 35644  df-padd 35936  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245  df-trl 36300

This theorem is referenced by:  cdlemk22-3  37043