Theorem cdlemk5u 31343

Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk1.b	|- B = ( Base ` K )
cdlemk1.l	|- .<_ = ( le ` K )
cdlemk1.j	|- .\/ = ( join ` K )
cdlemk1.m	|- ./\ = ( meet ` K )
cdlemk1.a	|- A = ( Atoms ` K )
cdlemk1.h	|- H = ( LHyp ` K )
cdlemk1.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r	|- R = ( ( trL ` K ) ` W )
cdlemk1.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o	|- O = ( S ` D )

Assertion

Ref

Expression

cdlemk5u

|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) )

Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   D, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i

Allowed substitution hints:   A( f)   B( f, i)   S( f, i)   G( f, i)   H( f)   K( f)   .<_ ( f)   O( f, i)   X( f, i)

Proof of Theorem cdlemk5u

Step	Hyp	Ref	Expression
1		cdlemk1.b	. 2 |- B = ( Base ` K )
2		cdlemk1.l	. 2 |- .<_ = ( le ` K )
3		simp11l 1068	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> K e. HL )
4		hllat 29846	. . 3 |- ( K e. HL -> K e. Lat )
5	3, 4	syl 16	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> K e. Lat )
6		simp22l 1076	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> P e. A )
7		simp1 957	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
8		simp211 1095	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> N e. T )
9		simp22 991	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
10		simp23 992	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` F ) = ( R ` N ) )
11	8, 9, 10	3jca 1134	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
12		simp3l1 1062	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> F =/= ( _I |` B ) )
13		simp3l2 1063	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> D =/= ( _I |` B ) )
14		simp3r1 1065	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
15	12, 13, 14	3jca 1134	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) )
16		cdlemk1.j	. . . . . . 7 |- .\/ = ( join ` K )
17		cdlemk1.m	. . . . . . 7 |- ./\ = ( meet ` K )
18		cdlemk1.a	. . . . . . 7 |- A = ( Atoms ` K )
19		cdlemk1.h	. . . . . . 7 |- H = ( LHyp ` K )
20		cdlemk1.t	. . . . . . 7 |- T = ( ( LTrn ` K ) ` W )
21		cdlemk1.r	. . . . . . 7 |- R = ( ( trL ` K ) ` W )
22		cdlemk1.s	. . . . . . 7 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
23		cdlemk1.o	. . . . . . 7 |- O = ( S ` D )
24	1, 2, 16, 17, 18, 19, 20, 21, 22, 23	cdlemkoatnle 31333	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) e. A /\ -. ( O ` P ) .<_ W ) )
25	24	simpld 446	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) e. A )
26	7, 11, 15, 25	syl3anc 1184	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( O ` P ) e. A )
27	1, 16, 18	hlatjcl 29849	. . . 4 |- ( ( K e. HL /\ P e. A /\ ( O ` P ) e. A ) -> ( P .\/ ( O ` P ) ) e. B )
28	3, 6, 26, 27	syl3anc 1184	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( O ` P ) ) e. B )
29		simp11 987	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( K e. HL /\ W e. H ) )
30		simp212 1096	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> G e. T )
31	2, 18, 19, 20	ltrnat 30622	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
32	29, 30, 6, 31	syl3anc 1184	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( G ` P ) e. A )
33		simp13 989	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> D e. T )
34		simp3r2 1066	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
35	18, 19, 20, 21	trlcocnvat 31206	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ D e. T ) /\ ( R ` G ) =/= ( R ` D ) ) -> ( R ` ( G o. `' D ) ) e. A )
36	29, 30, 33, 34, 35	syl121anc 1189	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( G o. `' D ) ) e. A )
37	1, 16, 18	hlatjcl 29849	. . . 4 |- ( ( K e. HL /\ ( G ` P ) e. A /\ ( R ` ( G o. `' D ) ) e. A ) -> ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B )
38	3, 32, 36, 37	syl3anc 1184	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B )
39	1, 17	latmcl 14435	. . 3 |- ( ( K e. Lat /\ ( P .\/ ( O ` P ) ) e. B /\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. B )
40	5, 28, 38, 39	syl3anc 1184	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. B )
41	2, 18, 19, 20	ltrnat 30622	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ D e. T /\ P e. A ) -> ( D ` P ) e. A )
42	29, 33, 6, 41	syl3anc 1184	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( D ` P ) e. A )
43	1, 18, 19, 20, 21	trlnidat 30655	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ D e. T /\ D =/= ( _I |` B ) ) -> ( R ` D ) e. A )
44	29, 33, 13, 43	syl3anc 1184	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` D ) e. A )
45	1, 16, 18	hlatjcl 29849	. . . 4 |- ( ( K e. HL /\ ( D ` P ) e. A /\ ( R ` D ) e. A ) -> ( ( D ` P ) .\/ ( R ` D ) ) e. B )
46	3, 42, 44, 45	syl3anc 1184	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( D ` P ) .\/ ( R ` D ) ) e. B )
47	1, 16, 18	hlatjcl 29849	. . . 4 |- ( ( K e. HL /\ ( D ` P ) e. A /\ ( R ` ( G o. `' D ) ) e. A ) -> ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B )
48	3, 42, 36, 47	syl3anc 1184	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B )
49	1, 17	latmcl 14435	. . 3 |- ( ( K e. Lat /\ ( ( D ` P ) .\/ ( R ` D ) ) e. B /\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B ) -> ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. B )
50	5, 46, 48, 49	syl3anc 1184	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. B )
51		simp213 1097	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> X e. T )
52	2, 18, 19, 20	ltrnat 30622	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A )
53	29, 51, 6, 52	syl3anc 1184	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( X ` P ) e. A )
54		simp3r3 1067	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` X ) =/= ( R ` D ) )
55	18, 19, 20, 21	trlcocnvat 31206	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ D e. T ) /\ ( R ` X ) =/= ( R ` D ) ) -> ( R ` ( X o. `' D ) ) e. A )
56	29, 51, 33, 54, 55	syl121anc 1189	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( R ` ( X o. `' D ) ) e. A )
57	1, 16, 18	hlatjcl 29849	. . 3 |- ( ( K e. HL /\ ( X ` P ) e. A /\ ( R ` ( X o. `' D ) ) e. A ) -> ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) e. B )
58	3, 53, 56, 57	syl3anc 1184	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) e. B )
59	1, 2, 16, 17, 18, 19, 20, 21, 22, 23	cdlemk1u 31341	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P .\/ ( O ` P ) ) .<_ ( ( D ` P ) .\/ ( R ` D ) ) )
60	7, 11, 15, 59	syl3anc 1184	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( P .\/ ( O ` P ) ) .<_ ( ( D ` P ) .\/ ( R ` D ) ) )
61	1, 2, 17	latmlem1 14465	. . . . 5 |- ( ( K e. Lat /\ ( ( P .\/ ( O ` P ) ) e. B /\ ( ( D ` P ) .\/ ( R ` D ) ) e. B /\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) e. B ) ) -> ( ( P .\/ ( O ` P ) ) .<_ ( ( D ` P ) .\/ ( R ` D ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) )
62	5, 28, 46, 38, 61	syl13anc 1186	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) .<_ ( ( D ` P ) .\/ ( R ` D ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) )
63	60, 62	mpd 15	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
64		simp11r 1069	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> W e. H )
65	1, 2, 16, 18, 19, 20, 21	cdlemk2 31314	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( D e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) = ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) )
66	3, 64, 33, 30, 9, 65	syl221anc 1195	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) = ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) )
67	66	oveq2d 6056	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) = ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
68	63, 67	breqtrd 4196	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) )
69		simp3l3 1064	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> G =/= ( _I |` B ) )
70	13, 69	jca 519	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
71	1, 2, 16, 18, 19, 20, 21, 17	cdlemk5a 31317	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( D e. T /\ G e. T /\ X e. T ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) )
72	3, 64, 33, 30, 51, 34, 70, 9, 71	syl233anc 1213	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( ( D ` P ) .\/ ( R ` D ) ) ./\ ( ( D ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) )
73	1, 2, 5, 40, 50, 58, 68, 72	lattrd 14442	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` X ) =/= ( R ` D ) ) ) ) -> ( ( P .\/ ( O ` P ) ) ./\ ( ( G ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ ( ( X ` P ) .\/ ( R ` ( X o. `' D ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   class class class wbr 4172   e. cmpt 4226   _I cid 4453   `'ccnv 4836   |` cres 4839   o. ccom 4841   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640

This theorem is referenced by:  cdlemk6u  31344

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641