Theorem cdlemk12 35655

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 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 ) ) ) -> K e. HL )
2		simp22l 1115	. 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 1026	. . 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 1028	. . 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 34945	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
10	3, 4, 2, 9	syl3anc 1228	. 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 1027	. . . 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 1114	. . . 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 1176	. . 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 1113	. . . 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 1030	. . . 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 1031	. . . 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 1176	. . 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 1143	. . 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 1145	. . 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 1122	. . 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 35651	. . 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 1240	. 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 1034	. . . 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 2738	. . 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 35529	. . 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 1233	. 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 996	. . 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 1144	. . 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 1121	. . 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 35651	. . 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 1240	. 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 35652	. . . . 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 1240	. . . 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 34169	. . . . . 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 34978	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
42	3, 4, 33, 41	syl3anc 1228	. . . . . 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 34172	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( P .\/ ( R ` G ) ) e. B )
44	1, 2, 42, 43	syl3anc 1228	. . . . 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 34945	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
46	3, 14, 2, 45	syl3anc 1228	. . . . . 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 35529	. . . . . . 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 1233	. . . . . 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 34172	. . . . . 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 1228	. . . . 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 15562	. . . . 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 1228	. . . 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 4467	. . 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 34971	. . . 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 1228	. . 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 4471	. 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 997	. . 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 1032	. . 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 2467	. . . 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 35654	. . 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 1240	. 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 34181	. . . . 5 |- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
63	1, 2, 42, 62	syl3anc 1228	. . . 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 4471	. . 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 35650	. . . . . 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 1240	. . . . 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 34944	. . . . 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 1228	. . . 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 34951	. . . . . . . 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 34970	. . . . . . . . 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 2760	. . . . . . 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 35533	. . . . . . 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 1237	. . . . . 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 2738	. . . . 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 35543	. . . . . . 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 1228	. . . . . 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 5869	. . . . 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 2770	. . . 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 35524	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ `' G e. T ) -> ( X o. `' G ) e. T )
82	3, 12, 70, 81	syl3anc 1228	. . . . 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 34989	. . . . 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 34989	. . . . 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 34838	. . . 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 1256	. . 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 4464	. . 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 34332	. 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 1260	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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4447   |-> cmpt 4505   _I cid 4790   `'ccnv 4998   |` cres 5001   o. ccom 5003   ` cfv 5587   iota_crio 6243  (class class class)co 6283   Basecbs 14489   lecple 14561   joincjn 15430   meetcmee 15431   Latclat 15531   Atomscatm 34069   HLchlt 34156   LHypclh 34789   LTrncltrn 34906   trLctrl 34963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-riotaBAD 33765

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-undef 7002  df-map 7422  df-poset 15432  df-plt 15444  df-lub 15460  df-glb 15461  df-join 15462  df-meet 15463  df-p0 15525  df-p1 15526  df-lat 15532  df-clat 15594  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304  df-lvols 34305  df-lines 34306  df-psubsp 34308  df-pmap 34309  df-padd 34601  df-lhyp 34793  df-laut 34794  df-ldil 34909  df-ltrn 34910  df-trl 34964

This theorem is referenced by:  cdlemk21N  35678  cdlemk20  35679