Theorem cdlemk12 34494

Description: Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
cdlemk12	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )

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

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

Proof of Theorem cdlemk12

Step	Hyp	Ref	Expression
1		simp11l 1099	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> K e. HL )
2		simp22l 1107	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> P e. A )
3		simp11 1018	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( K e. HL /\ W e. H ) )
4		simp13 1020	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> G e. T )
5		cdlemk.l	. . . 4 |- .<_ = ( le ` K )
6		cdlemk.a	. . . 4 |- A = ( Atoms ` K )
7		cdlemk.h	. . . 4 |- H = ( LHyp ` K )
8		cdlemk.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
9	5, 6, 7, 8	ltrnat 33784	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
10	3, 4, 2, 9	syl3anc 1218	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( G ` P ) e. A )
11		simp12 1019	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> F e. T )
12		simp21r 1106	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> X e. T )
13	3, 11, 12	3jca 1168	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ X e. T ) )
14		simp21l 1105	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> N e. T )
15		simp22 1022	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( P e. A /\ -. P .<_ W ) )
16		simp23 1023	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` F ) = ( R ` N ) )
17	14, 15, 16	3jca 1168	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
18		simp311 1135	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> F =/= ( _I |` B ) )
19		simp313 1137	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> X =/= ( _I |` B ) )
20		simp32r 1114	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` X ) =/= ( R ` F ) )
21		cdlemk.b	. . . 4 |- B = ( Base ` K )
22		cdlemk.j	. . . 4 |- .\/ = ( join ` K )
23		cdlemk.r	. . . 4 |- R = ( ( trL ` K ) ` W )
24		cdlemk.m	. . . 4 |- ./\ = ( meet ` K )
25		cdlemk.s	. . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
26	21, 5, 22, 6, 7, 8, 23, 24, 25	cdlemksat 34490	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ X e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` X ) ` P ) e. A )
27	13, 17, 18, 19, 20, 26	syl113anc 1230	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` X ) ` P ) e. A )
28		simp33 1026	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` G ) =/= ( R ` X ) )
29	28	necomd 2695	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` X ) =/= ( R ` G ) )
30	6, 7, 8, 23	trlcocnvat 34368	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. T /\ G e. T ) /\ ( R ` X ) =/= ( R ` G ) ) -> ( R ` ( X o. `' G ) ) e. A )
31	3, 12, 4, 29, 30	syl121anc 1223	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` ( X o. `' G ) ) e. A )
32		simp1 988	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) )
33		simp312 1136	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> G =/= ( _I |` B ) )
34		simp32l 1113	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` G ) =/= ( R ` F ) )
35	21, 5, 22, 6, 7, 8, 23, 24, 25	cdlemksat 34490	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) e. A )
36	32, 17, 18, 33, 34, 35	syl113anc 1230	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) e. A )
37	21, 5, 22, 6, 7, 8, 23, 24, 25	cdlemksv2 34491	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
38	32, 17, 18, 33, 34, 37	syl113anc 1230	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
39		hllat 33008	. . . . . 6 |- ( K e. HL -> K e. Lat )
40	1, 39	syl 16	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> K e. Lat )
41	21, 6, 7, 8, 23	trlnidat 33817	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
42	3, 4, 33, 41	syl3anc 1218	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` G ) e. A )
43	21, 22, 6	hlatjcl 33011	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( P .\/ ( R ` G ) ) e. B )
44	1, 2, 42, 43	syl3anc 1218	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( P .\/ ( R ` G ) ) e. B )
45	5, 6, 7, 8	ltrnat 33784	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
46	3, 14, 2, 45	syl3anc 1218	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( N ` P ) e. A )
47	6, 7, 8, 23	trlcocnvat 34368	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ F e. T ) /\ ( R ` G ) =/= ( R ` F ) ) -> ( R ` ( G o. `' F ) ) e. A )
48	3, 4, 11, 34, 47	syl121anc 1223	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` ( G o. `' F ) ) e. A )
49	21, 22, 6	hlatjcl 33011	. . . . . 6 |- ( ( K e. HL /\ ( N ` P ) e. A /\ ( R ` ( G o. `' F ) ) e. A ) -> ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) e. B )
50	1, 46, 48, 49	syl3anc 1218	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) e. B )
51	21, 5, 24	latmle1 15246	. . . . 5 |- ( ( K e. Lat /\ ( P .\/ ( R ` G ) ) e. B /\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) e. B ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( P .\/ ( R ` G ) ) )
52	40, 44, 50, 51	syl3anc 1218	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) .<_ ( P .\/ ( R ` G ) ) )
53	38, 52	eqbrtrd 4312	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) .<_ ( P .\/ ( R ` G ) ) )
54	5, 22, 6, 7, 8, 23	trljat1 33810	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
55	3, 4, 15, 54	syl3anc 1218	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) )
56	53, 55	breqtrd 4316	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) .<_ ( P .\/ ( G ` P ) ) )
57		simp2 989	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
58		simp31 1024	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) )
59		eqid 2443	. . . 4 |- ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) ) = ( ( ( G ` P ) .\/ ( X ` P ) ) ./\ ( ( R ` ( G o. `' F ) ) .\/ ( R ` ( X o. `' F ) ) ) )
60	21, 5, 22, 6, 7, 8, 23, 24, 25, 59	cdlemk11 34493	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
61	32, 57, 58, 34, 20, 60	syl113anc 1230	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
62	5, 22, 6	hlatlej2 33020	. . . . 5 |- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
63	1, 2, 42, 62	syl3anc 1218	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
64	63, 55	breqtrd 4316	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` G ) .<_ ( P .\/ ( G ` P ) ) )
65	21, 5, 22, 6, 7, 8, 23, 24, 25	cdlemksel 34489	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ X e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ X =/= ( _I |` B ) /\ ( R ` X ) =/= ( R ` F ) ) ) -> ( S ` X ) e. T )
66	13, 17, 18, 19, 20, 65	syl113anc 1230	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( S ` X ) e. T )
67	5, 6, 7, 8	ltrnel 33783	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( S ` X ) e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( S ` X ) ` P ) e. A /\ -. ( ( S ` X ) ` P ) .<_ W ) )
68	3, 66, 15, 67	syl3anc 1218	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( ( S ` X ) ` P ) e. A /\ -. ( ( S ` X ) ` P ) .<_ W ) )
69	7, 8	ltrncnv 33790	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> `' G e. T )
70	3, 4, 69	syl2anc 661	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> `' G e. T )
71	7, 8, 23	trlcnv 33809	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` `' G ) = ( R ` G ) )
72	3, 4, 71	syl2anc 661	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` `' G ) = ( R ` G ) )
73	72, 28	eqnetrd 2626	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` `' G ) =/= ( R ` X ) )
74	21, 7, 8, 23	trlcone 34372	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( `' G e. T /\ X e. T ) /\ ( ( R ` `' G ) =/= ( R ` X ) /\ X =/= ( _I |` B ) ) ) -> ( R ` `' G ) =/= ( R ` ( `' G o. X ) ) )
75	3, 70, 12, 73, 19, 74	syl122anc 1227	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` `' G ) =/= ( R ` ( `' G o. X ) ) )
76	75	necomd 2695	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` ( `' G o. X ) ) =/= ( R ` `' G ) )
77	7, 8	ltrncom 34382	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ `' G e. T /\ X e. T ) -> ( `' G o. X ) = ( X o. `' G ) )
78	3, 70, 12, 77	syl3anc 1218	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( `' G o. X ) = ( X o. `' G ) )
79	78	fveq2d 5695	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` ( `' G o. X ) ) = ( R ` ( X o. `' G ) ) )
80	76, 79, 72	3netr3d 2634	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` ( X o. `' G ) ) =/= ( R ` G ) )
81	7, 8	ltrnco 34363	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ `' G e. T ) -> ( X o. `' G ) e. T )
82	3, 12, 70, 81	syl3anc 1218	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( X o. `' G ) e. T )
83	5, 7, 8, 23	trlle 33828	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X o. `' G ) e. T ) -> ( R ` ( X o. `' G ) ) .<_ W )
84	3, 82, 83	syl2anc 661	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` ( X o. `' G ) ) .<_ W )
85	5, 7, 8, 23	trlle 33828	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
86	3, 4, 85	syl2anc 661	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( R ` G ) .<_ W )
87	5, 22, 6, 7	lhp2atnle 33677	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( ( S ` X ) ` P ) e. A /\ -. ( ( S ` X ) ` P ) .<_ W ) /\ ( R ` ( X o. `' G ) ) =/= ( R ` G ) ) /\ ( ( R ` ( X o. `' G ) ) e. A /\ ( R ` ( X o. `' G ) ) .<_ W ) /\ ( ( R ` G ) e. A /\ ( R ` G ) .<_ W ) ) -> -. ( R ` G ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
88	3, 68, 80, 31, 84, 42, 86, 87	syl322anc 1246	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> -. ( R ` G ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
89		nbrne1 4309	. . 3 |- ( ( ( R ` G ) .<_ ( P .\/ ( G ` P ) ) /\ -. ( R ` G ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) -> ( P .\/ ( G ` P ) ) =/= ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
90	64, 88, 89	syl2anc 661	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( P .\/ ( G ` P ) ) =/= ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) )
91	5, 22, 24, 6	2atm 33171	. 2 |- ( ( ( K e. HL /\ P e. A /\ ( G ` P ) e. A ) /\ ( ( ( S ` X ) ` P ) e. A /\ ( R ` ( X o. `' G ) ) e. A /\ ( ( S ` G ) ` P ) e. A ) /\ ( ( ( S ` G ) ` P ) .<_ ( P .\/ ( G ` P ) ) /\ ( ( S ` G ) ` P ) .<_ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) /\ ( P .\/ ( G ` P ) ) =/= ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )
92	1, 2, 10, 27, 31, 36, 56, 61, 90, 91	syl333anc 1250	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ X =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` X ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` X ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2606   class class class wbr 4292   e. cmpt 4350   _I cid 4631   `'ccnv 4839   |` cres 4842   o. ccom 4844   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   Latclat 15215   Atomscatm 32908   HLchlt 32995   LHypclh 33628   LTrncltrn 33745   trLctrl 33802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-riotaBAD 32604

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-undef 6792  df-map 7216  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803

This theorem is referenced by:  cdlemk21N  34517  cdlemk20  34518