Theorem cdlemkid4 36222

Description: Lemma for cdlemkid 36224. (Contributed by NM, 25-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
cdlemkid4	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )

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

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

Proof of Theorem cdlemkid4

Step	Hyp	Ref	Expression
1		simp3r 1090	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> G = ( _I |` B ) )
2		cdlemk5.b	. . . . . 6 |- B = ( Base ` K )
3		cdlemk5.h	. . . . . 6 |- H = ( LHyp ` K )
4		cdlemk5.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
5	2, 3, 4	idltrn 35436	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( _I |` B ) e. T )
6	5	3ad2ant1 1082	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> ( _I |` B ) e. T )
7	1, 6	eqeltrd 2701	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> G e. T )
8		cdlemk5.x	. . . . . . 7 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
9	8	csbeq2i 3993	. . . . . 6 |- [_ G / g ]_ X = [_ G / g ]_ ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
10		csbriota 6623	. . . . . 6 |- [_ G / g ]_ ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) = ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
11	9, 10	eqtri 2644	. . . . 5 |- [_ G / g ]_ X = ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) )
12	11	a1i 11	. . . 4 |- ( G e. T -> [_ G / g ]_ X = ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) )
13		sbcralg 3513	. . . . . 6 |- ( G e. T -> ( [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> A. b e. T [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) )
14		sbcimg 3477	. . . . . . . 8 |- ( G e. T -> ( [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> [. G / g ]. ( z ` P ) = Y ) ) )
15		sbc3an 3494	. . . . . . . . . 10 |- ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) <-> ( [. G / g ]. b =/= ( _I |` B ) /\ [. G / g ]. ( R ` b ) =/= ( R ` F ) /\ [. G / g ]. ( R ` b ) =/= ( R ` g ) ) )
16		sbcg 3503	. . . . . . . . . . 11 |- ( G e. T -> ( [. G / g ]. b =/= ( _I |` B ) <-> b =/= ( _I |` B ) ) )
17		sbcg 3503	. . . . . . . . . . 11 |- ( G e. T -> ( [. G / g ]. ( R ` b ) =/= ( R ` F ) <-> ( R ` b ) =/= ( R ` F ) ) )
18		sbcne12 3986	. . . . . . . . . . . 12 |- ( [. G / g ]. ( R ` b ) =/= ( R ` g ) <-> [_ G / g ]_ ( R ` b ) =/= [_ G / g ]_ ( R ` g ) )
19		csbconstg 3546	. . . . . . . . . . . . 13 |- ( G e. T -> [_ G / g ]_ ( R ` b ) = ( R ` b ) )
20		csbfv 6233	. . . . . . . . . . . . . 14 |- [_ G / g ]_ ( R ` g ) = ( R ` G )
21	20	a1i 11	. . . . . . . . . . . . 13 |- ( G e. T -> [_ G / g ]_ ( R ` g ) = ( R ` G ) )
22	19, 21	neeq12d 2855	. . . . . . . . . . . 12 |- ( G e. T -> ( [_ G / g ]_ ( R ` b ) =/= [_ G / g ]_ ( R ` g ) <-> ( R ` b ) =/= ( R ` G ) ) )
23	18, 22	syl5bb 272	. . . . . . . . . . 11 |- ( G e. T -> ( [. G / g ]. ( R ` b ) =/= ( R ` g ) <-> ( R ` b ) =/= ( R ` G ) ) )
24	16, 17, 23	3anbi123d 1399	. . . . . . . . . 10 |- ( G e. T -> ( ( [. G / g ]. b =/= ( _I |` B ) /\ [. G / g ]. ( R ` b ) =/= ( R ` F ) /\ [. G / g ]. ( R ` b ) =/= ( R ` g ) ) <-> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
25	15, 24	syl5bb 272	. . . . . . . . 9 |- ( G e. T -> ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) <-> ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) )
26		sbceq2g 3990	. . . . . . . . 9 |- ( G e. T -> ( [. G / g ]. ( z ` P ) = Y <-> ( z ` P ) = [_ G / g ]_ Y ) )
27	25, 26	imbi12d 334	. . . . . . . 8 |- ( G e. T -> ( ( [. G / g ]. ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> [. G / g ]. ( z ` P ) = Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
28	14, 27	bitrd 268	. . . . . . 7 |- ( G e. T -> ( [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
29	28	ralbidv 2986	. . . . . 6 |- ( G e. T -> ( A. b e. T [. G / g ]. ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
30	13, 29	bitrd 268	. . . . 5 |- ( G e. T -> ( [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
31	30	riotabidv 6613	. . . 4 |- ( G e. T -> ( iota_ z e. T [. G / g ]. A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
32	12, 31	eqtrd 2656	. . 3 |- ( G e. T -> [_ G / g ]_ X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
33	7, 32	syl 17	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) )
34		simpl1 1064	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
35		simpl2 1065	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) )
36		simpl3l 1116	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
37		simpl3r 1117	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> G = ( _I |` B ) )
38		simprlr 803	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b e. T )
39		simprr1 1109	. . . . . . . . . . 11 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> b =/= ( _I |` B ) )
40	38, 39	jca 554	. . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( b e. T /\ b =/= ( _I |` B ) ) )
41		cdlemk5.l	. . . . . . . . . . 11 |- .<_ = ( le ` K )
42		cdlemk5.j	. . . . . . . . . . 11 |- .\/ = ( join ` K )
43		cdlemk5.m	. . . . . . . . . . 11 |- ./\ = ( meet ` K )
44		cdlemk5.a	. . . . . . . . . . 11 |- A = ( Atoms ` K )
45		cdlemk5.r	. . . . . . . . . . 11 |- R = ( ( trL ` K ) ` W )
46		cdlemk5.z	. . . . . . . . . . 11 |- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
47		cdlemk5.y	. . . . . . . . . . 11 |- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
48	2, 41, 42, 43, 44, 3, 4, 45, 46, 47	cdlemkid2 36212	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) /\ ( b e. T /\ b =/= ( _I |` B ) ) ) ) -> [_ G / g ]_ Y = P )
49	34, 35, 36, 37, 40, 48	syl113anc 1338	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> [_ G / g ]_ Y = P )
50	49	eqeq2d 2632	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> ( z ` P ) = P ) )
51		simprll 802	. . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> z e. T )
52	2, 41, 44, 3, 4	ltrnideq 35462	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ z e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( z = ( _I |` B ) <-> ( z ` P ) = P ) )
53	34, 51, 36, 52	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( z = ( _I |` B ) <-> ( z ` P ) = P ) )
54	50, 53	bitr4d 271	. . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ ( ( z e. T /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> z = ( _I |` B ) ) )
55	54	exp44 641	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> ( z e. T -> ( b e. T -> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> z = ( _I |` B ) ) ) ) ) )
56	55	imp41 619	. . . . 5 |- ( ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ z e. T ) /\ b e. T ) /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) ) -> ( ( z ` P ) = [_ G / g ]_ Y <-> z = ( _I |` B ) ) )
57	56	pm5.74da 723	. . . 4 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ z e. T ) /\ b e. T ) -> ( ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) <-> ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
58	57	ralbidva 2985	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) /\ z e. T ) -> ( A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) <-> A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
59	58	riotabidva 6627	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = [_ G / g ]_ Y ) ) = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )
60	33, 59	eqtrd 2656	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T /\ ( R ` F ) = ( R ` N ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ G = ( _I |` B ) ) ) -> [_ G / g ]_ X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> z = ( _I |` B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   [.wsbc 3435   [_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:  cdlemkid5  36223  cdlemkid  36224