Theorem cdlemk47 36237

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 2, p. 120. G, I stand for g, h. X represents tau. (Contributed by NM, 22-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk5.b	|- B = ( Base ` K )
cdlemk5.l	|- .<_ = ( le ` K )
cdlemk5.j	|- .\/ = ( join ` K )
cdlemk5.m	|- ./\ = ( meet ` K )
cdlemk5.a	|- A = ( Atoms ` K )
cdlemk5.h	|- H = ( LHyp ` K )
cdlemk5.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r	|- R = ( ( trL ` K ) ` W )
cdlemk5.z	|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y	|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )

Assertion

Ref	Expression
cdlemk47	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) = ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )

Distinct variable groups:   ./\ , g   .\/ , g   B, g   P, g   R, g   T, g   g, Z   g, b, G, z   ./\ , b, z   .<_ , b   z, g, .<_   .\/ , b, z   A, b, g, z   B, b, z   F, b, g, z   z, G   H, b, g, z   K, b, g, z   N, b, g, z   P, b, z   R, b, z   T, b, z   W, b, g, z   z, Y   G, b   I, b, g, z

Allowed substitution hints:   X( z, g, b)   Y( g, b)   Z( z, b)

Proof of Theorem cdlemk47

Step	Hyp	Ref	Expression
1		simp11l 1172	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> K e. HL )
2		simp11 1091	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( K e. HL /\ W e. H ) )
3		simp12 1092	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
4		simp13 1093	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G e. T /\ G =/= ( _I |` B ) ) )
5		simp21 1094	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> N e. T )
6		simp22 1095	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( P e. A /\ -. P .<_ W ) )
7		simp23 1096	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` F ) = ( R ` N ) )
8		cdlemk5.b	. . . . . 6 |- B = ( Base ` K )
9		cdlemk5.l	. . . . . 6 |- .<_ = ( le ` K )
10		cdlemk5.j	. . . . . 6 |- .\/ = ( join ` K )
11		cdlemk5.m	. . . . . 6 |- ./\ = ( meet ` K )
12		cdlemk5.a	. . . . . 6 |- A = ( Atoms ` K )
13		cdlemk5.h	. . . . . 6 |- H = ( LHyp ` K )
14		cdlemk5.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
15		cdlemk5.r	. . . . . 6 |- R = ( ( trL ` K ) ` W )
16		cdlemk5.z	. . . . . 6 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
17		cdlemk5.y	. . . . . 6 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
18		cdlemk5.x	. . . . . 6 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
19	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemk35s 36225	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ G / g ]_ X e. T )
20	2, 3, 4, 5, 6, 7, 19	syl132anc 1344	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ G / g ]_ X e. T )
21	9, 12, 13, 14	ltrnel 35425	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( [_ G / g ]_ X ` P ) e. A /\ -. ( [_ G / g ]_ X ` P ) .<_ W ) )
22	2, 20, 6, 21	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X ` P ) e. A /\ -. ( [_ G / g ]_ X ` P ) .<_ W ) )
23	22	simpld 475	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ G / g ]_ X ` P ) e. A )
24		simp31 1097	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> I e. T )
25		simp32 1098	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> I =/= ( _I |` B ) )
26	8, 12, 13, 14, 15	trlnidat 35460	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ I e. T /\ I =/= ( _I |` B ) ) -> ( R ` I ) e. A )
27	2, 24, 25, 26	syl3anc 1326	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` I ) e. A )
28	24, 25	jca 554	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) ) )
29	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemk35s 36225	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ I / g ]_ X e. T )
30	2, 3, 28, 5, 6, 7, 29	syl132anc 1344	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ I / g ]_ X e. T )
31		simp22l 1180	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> P e. A )
32	9, 12, 13, 14	ltrnat 35426	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ [_ I / g ]_ X e. T /\ P e. A ) -> ( [_ I / g ]_ X ` P ) e. A )
33	2, 30, 31, 32	syl3anc 1326	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ I / g ]_ X ` P ) e. A )
34		simp13l 1176	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> G e. T )
35		simp13r 1177	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> G =/= ( _I |` B ) )
36	8, 12, 13, 14, 15	trlnidat 35460	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
37	2, 34, 35, 36	syl3anc 1326	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` G ) e. A )
38	13, 14	ltrnco 36007	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ I e. T ) -> ( G o. I ) e. T )
39	2, 34, 24, 38	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G o. I ) e. T )
40	34, 24	jca 554	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G e. T /\ I e. T ) )
41		simp33 1099	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` G ) =/= ( R ` I ) )
42	8, 13, 14, 15	trlconid 36013	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ I e. T ) /\ ( R ` G ) =/= ( R ` I ) ) -> ( G o. I ) =/= ( _I |` B ) )
43	2, 40, 41, 42	syl3anc 1326	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( G o. I ) =/= ( _I |` B ) )
44	39, 43	jca 554	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( G o. I ) e. T /\ ( G o. I ) =/= ( _I |` B ) ) )
45	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemk35s 36225	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( ( G o. I ) e. T /\ ( G o. I ) =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ ( G o. I ) / g ]_ X e. T )
46	2, 3, 44, 5, 6, 7, 45	syl132anc 1344	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> [_ ( G o. I ) / g ]_ X e. T )
47	9, 12, 13, 14	ltrnat 35426	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ [_ ( G o. I ) / g ]_ X e. T /\ P e. A ) -> ( [_ ( G o. I ) / g ]_ X ` P ) e. A )
48	2, 46, 31, 47	syl3anc 1326	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) e. A )
49	24, 25, 43	3jca 1242	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) /\ ( G o. I ) =/= ( _I |` B ) ) )
50	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemk46 36236	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( G o. I ) =/= ( _I |` B ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) .<_ ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) )
51	49, 50	syld3an3 1371	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) .<_ ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) )
52	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemk45 36235	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( G o. I ) =/= ( _I |` B ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) )
53	49, 52	syld3an3 1371	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) )
54	9, 13, 14, 15	trlle 35471	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ I e. T ) -> ( R ` I ) .<_ W )
55	2, 24, 54	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` I ) .<_ W )
56	27, 55	jca 554	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( R ` I ) e. A /\ ( R ` I ) .<_ W ) )
57	9, 13, 14, 15	trlle 35471	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
58	2, 34, 57	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` G ) .<_ W )
59	37, 58	jca 554	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( R ` G ) e. A /\ ( R ` G ) .<_ W ) )
60	41	necomd 2849	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( R ` I ) =/= ( R ` G ) )
61	9, 10, 12, 13	lhp2atne 35320	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( [_ G / g ]_ X ` P ) e. A /\ -. ( [_ G / g ]_ X ` P ) .<_ W ) /\ ( [_ I / g ]_ X ` P ) e. A ) /\ ( ( ( R ` I ) e. A /\ ( R ` I ) .<_ W ) /\ ( ( R ` G ) e. A /\ ( R ` G ) .<_ W ) ) /\ ( R ` I ) =/= ( R ` G ) ) -> ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) =/= ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) )
62	2, 22, 33, 56, 59, 60, 61	syl321anc 1348	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) =/= ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) )
63	9, 10, 11, 12	2atm 34813	. 2 |- ( ( ( K e. HL /\ ( [_ G / g ]_ X ` P ) e. A /\ ( R ` I ) e. A ) /\ ( ( [_ I / g ]_ X ` P ) e. A /\ ( R ` G ) e. A /\ ( [_ ( G o. I ) / g ]_ X ` P ) e. A ) /\ ( ( [_ ( G o. I ) / g ]_ X ` P ) .<_ ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) /\ ( [_ ( G o. I ) / g ]_ X ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) /\ ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) =/= ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) = ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )
64	1, 23, 27, 33, 37, 48, 51, 53, 62, 63	syl333anc 1358	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` I ) ) ) -> ( [_ ( G o. I ) / g ]_ X ` P ) = ( ( ( [_ G / g ]_ X ` P ) .\/ ( R ` I ) ) ./\ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` G ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [_csb 3533   class class class wbr 4653   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemk52  36242