Theorem cdlemk47 33968

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 1108	. 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 1027	. . . 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 1028	. . . . 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 1029	. . . . 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 1030	. . . . 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 1031	. . . . 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 1032	. . . . 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 33956	. . . . 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 1248	. . . 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 33156	. . . 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 1230	. . 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 457	. 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 1033	. . 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 1034	. . 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 33191	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ I e. T /\ I =/= ( _I |` B ) ) -> ( R ` I ) e. A )
27	2, 24, 25, 26	syl3anc 1230	. 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 530	. . . 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 33956	. . . 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 1248	. . 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 1116	. . 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 33157	. . 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 1230	. 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 1112	. . 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 1113	. . 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 33191	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
37	2, 34, 35, 36	syl3anc 1230	. 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 33738	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ I e. T ) -> ( G o. I ) e. T )
39	2, 34, 24, 38	syl3anc 1230	. . . . 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 530	. . . . . 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 1035	. . . . . 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 33744	. . . . . 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 1230	. . . . 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 530	. . . 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 33956	. . . 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 1248	. . 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 33157	. . 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 1230	. 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 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 ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) /\ ( G o. I ) =/= ( _I |` B ) ) )
50	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	cdlemk46 33967	. . 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 1275	. 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 33966	. . 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 1275	. 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 33202	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ I e. T ) -> ( R ` I ) .<_ W )
55	2, 24, 54	syl2anc 659	. . . 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 530	. . 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 33202	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
58	2, 34, 57	syl2anc 659	. . . 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 530	. . 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 2674	. . 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 33051	. . 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 1252	. 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 32544	. 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 1262	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 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2754   [_csb 3373   class class class wbr 4395   _I cid 4733   `'ccnv 4822   |` cres 4825   o. ccom 4827   ` cfv 5569   iota_crio 6239  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Atomscatm 32281   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   trLctrl 33176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-riotaBAD 31977

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-undef 7005  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177

This theorem is referenced by:  cdlemk52  33973