Theorem cdlemk12 31332

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 1068	. 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 1076	. 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 987	. . 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 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 ) ) ) -> 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 30622	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
10	3, 4, 2, 9	syl3anc 1184	. 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 988	. . . 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 1075	. . . 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 1134	. . 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 1074	. . . 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 991	. . . 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 992	. . . 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 1134	. . 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 1104	. . 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 1106	. . 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 1083	. . 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 31328	. . 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 1196	. 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 995	. . . 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 2650	. . 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 31206	. . 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 1189	. 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 957	. . 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 1105	. . 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 1082	. . 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 31328	. . 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 1196	. 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 31329	. . . . 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 1196	. . . 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 29846	. . . . . 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 30655	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
42	3, 4, 33, 41	syl3anc 1184	. . . . . 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 29849	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( P .\/ ( R ` G ) ) e. B )
44	1, 2, 42, 43	syl3anc 1184	. . . . 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 30622	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
46	3, 14, 2, 45	syl3anc 1184	. . . . . 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 31206	. . . . . . 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 1189	. . . . . 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 29849	. . . . . 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 1184	. . . . 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 14460	. . . . 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 1184	. . . 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 4192	. . 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 30648	. . . 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 1184	. . 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 4196	. 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 958	. . 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 993	. . 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 2404	. . . 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 31331	. . 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 1196	. 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 29858	. . . . 5 |- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
63	1, 2, 42, 62	syl3anc 1184	. . . 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 4196	. . 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 31327	. . . . . 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 1196	. . . . 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 30621	. . . . 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 1184	. . . 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 30628	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> `' G e. T )
70	3, 4, 69	syl2anc 643	. . . . . . 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 30647	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` `' G ) = ( R ` G ) )
72	3, 4, 71	syl2anc 643	. . . . . . . 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 2585	. . . . . . 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 31210	. . . . . . 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 1193	. . . . . 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 2650	. . . . 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 31220	. . . . . . 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 1184	. . . . . 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 5691	. . . . 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 2593	. . . 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 31201	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ `' G e. T ) -> ( X o. `' G ) e. T )
82	3, 12, 70, 81	syl3anc 1184	. . . . 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 30666	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( X o. `' G ) e. T ) -> ( R ` ( X o. `' G ) ) .<_ W )
84	3, 82, 83	syl2anc 643	. . . 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 30666	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
86	3, 4, 85	syl2anc 643	. . . 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 30515	. . . 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 1212	. . 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 4189	. . 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 643	. 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 30009	. 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 1216	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 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:  cdlemk21N  31355  cdlemk20  31356

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