Theorem efgrelexlemb 18163

Description: If two words A , B are related under the free group equivalence, then there exist two extension sequences a , b such that a ends at A, b ends at B, and a and B have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )
efgval2.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
efgred.d	|- D = ( W \ U_ x e. W ran ( T ` x ) )
efgred.s	|- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
efgrelexlem.1	|- L = { <. i , j >. | E. c e. ( `' S " { i } ) E. d e. ( `' S " { j } ) ( c ` 0 ) = ( d ` 0 ) }

Assertion

Ref	Expression
efgrelexlemb	|- .~ C_ L

Distinct variable groups:   c, d, i, j   y, z   n, c, t, v, w, y, z, m, x   M, c   i, m, n, t, v, w, x, M, j   k, c, T, i, j, m, t, x   W, c   k, d, m, n, t, v, w, x, y, z, W, i, j   .~ , c, d, i, j, m, t, x, y, z   S, c, d, i, j   I, c, i, j, m, n, t, v, w, x, y, z   D, c, d, i, j, m, t

Allowed substitution hints:   D( x, y, z, w, v, k, n)   .~ ( w, v, k, n)   S( x, y, z, w, v, t, k, m, n)   T( y, z, w, v, n, d)   I( k, d)   L( x, y, z, w, v, t, i, j, k, m, n, c, d)   M( y, z, k, d)

Proof of Theorem efgrelexlemb

Dummy variables a b f g h r s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . 3 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . 3 |- .~ = ( ~FG ` I )
3		efgval2.m	. . 3 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . 3 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5	1, 2, 3, 4	efgval2 18137	. 2 |- .~ = |^| { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) }
6		efgrelexlem.1	. . . . . . . 8 |- L = { <. i , j >. | E. c e. ( `' S " { i } ) E. d e. ( `' S " { j } ) ( c ` 0 ) = ( d ` 0 ) }
7	6	relopabi 5245	. . . . . . 7 |- Rel L
8	7	a1i 11	. . . . . 6 |- ( T. -> Rel L )
9		simpr 477	. . . . . . 7 |- ( ( T. /\ f L g ) -> f L g )
10		eqcom 2629	. . . . . . . . . 10 |- ( ( a ` 0 ) = ( b ` 0 ) <-> ( b ` 0 ) = ( a ` 0 ) )
11	10	2rexbii 3042	. . . . . . . . 9 |- ( E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) <-> E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( b ` 0 ) = ( a ` 0 ) )
12		rexcom 3099	. . . . . . . . 9 |- ( E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( b ` 0 ) = ( a ` 0 ) <-> E. b e. ( `' S " { g } ) E. a e. ( `' S " { f } ) ( b ` 0 ) = ( a ` 0 ) )
13	11, 12	bitri 264	. . . . . . . 8 |- ( E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) <-> E. b e. ( `' S " { g } ) E. a e. ( `' S " { f } ) ( b ` 0 ) = ( a ` 0 ) )
14		efgred.d	. . . . . . . . 9 |- D = ( W \ U_ x e. W ran ( T ` x ) )
15		efgred.s	. . . . . . . . 9 |- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
16	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18162	. . . . . . . 8 |- ( f L g <-> E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) )
17	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18162	. . . . . . . 8 |- ( g L f <-> E. b e. ( `' S " { g } ) E. a e. ( `' S " { f } ) ( b ` 0 ) = ( a ` 0 ) )
18	13, 16, 17	3bitr4i 292	. . . . . . 7 |- ( f L g <-> g L f )
19	9, 18	sylib 208	. . . . . 6 |- ( ( T. /\ f L g ) -> g L f )
20	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18162	. . . . . . . . 9 |- ( g L h <-> E. r e. ( `' S " { g } ) E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) )
21		reeanv 3107	. . . . . . . . . 10 |- ( E. a e. ( `' S " { f } ) E. r e. ( `' S " { g } ) ( E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) /\ E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) ) <-> ( E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) /\ E. r e. ( `' S " { g } ) E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) ) )
22	1, 2, 3, 4, 14, 15	efgsfo 18152	. . . . . . . . . . . . . . . . . . . 20 |- S : dom S -onto-> W
23		fofn 6117	. . . . . . . . . . . . . . . . . . . 20 |- ( S : dom S -onto-> W -> S Fn dom S )
24	22, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- S Fn dom S
25		fniniseg 6338	. . . . . . . . . . . . . . . . . . 19 |- ( S Fn dom S -> ( r e. ( `' S " { g } ) <-> ( r e. dom S /\ ( S ` r ) = g ) ) )
26	24, 25	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( r e. ( `' S " { g } ) <-> ( r e. dom S /\ ( S ` r ) = g ) )
27		fniniseg 6338	. . . . . . . . . . . . . . . . . . 19 |- ( S Fn dom S -> ( b e. ( `' S " { g } ) <-> ( b e. dom S /\ ( S ` b ) = g ) ) )
28	24, 27	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- ( b e. ( `' S " { g } ) <-> ( b e. dom S /\ ( S ` b ) = g ) )
29		eqtr3 2643	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( S ` r ) = g /\ ( S ` b ) = g ) -> ( S ` r ) = ( S ` b ) )
30	1, 2, 3, 4, 14, 15	efgred 18161	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( r e. dom S /\ b e. dom S /\ ( S ` r ) = ( S ` b ) ) -> ( r ` 0 ) = ( b ` 0 ) )
31	30	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( r e. dom S /\ b e. dom S /\ ( S ` r ) = ( S ` b ) ) -> ( b ` 0 ) = ( r ` 0 ) )
32	31	3expa 1265	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( r e. dom S /\ b e. dom S ) /\ ( S ` r ) = ( S ` b ) ) -> ( b ` 0 ) = ( r ` 0 ) )
33	29, 32	sylan2 491	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( r e. dom S /\ b e. dom S ) /\ ( ( S ` r ) = g /\ ( S ` b ) = g ) ) -> ( b ` 0 ) = ( r ` 0 ) )
34	33	an4s 869	. . . . . . . . . . . . . . . . . 18 |- ( ( ( r e. dom S /\ ( S ` r ) = g ) /\ ( b e. dom S /\ ( S ` b ) = g ) ) -> ( b ` 0 ) = ( r ` 0 ) )
35	26, 28, 34	syl2anb 496	. . . . . . . . . . . . . . . . 17 |- ( ( r e. ( `' S " { g } ) /\ b e. ( `' S " { g } ) ) -> ( b ` 0 ) = ( r ` 0 ) )
36		eqeq2 2633	. . . . . . . . . . . . . . . . 17 |- ( ( r ` 0 ) = ( s ` 0 ) -> ( ( b ` 0 ) = ( r ` 0 ) <-> ( b ` 0 ) = ( s ` 0 ) ) )
37	35, 36	syl5ibcom 235	. . . . . . . . . . . . . . . 16 |- ( ( r e. ( `' S " { g } ) /\ b e. ( `' S " { g } ) ) -> ( ( r ` 0 ) = ( s ` 0 ) -> ( b ` 0 ) = ( s ` 0 ) ) )
38	37	reximdv 3016	. . . . . . . . . . . . . . 15 |- ( ( r e. ( `' S " { g } ) /\ b e. ( `' S " { g } ) ) -> ( E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) -> E. s e. ( `' S " { h } ) ( b ` 0 ) = ( s ` 0 ) ) )
39		eqeq1 2626	. . . . . . . . . . . . . . . . 17 |- ( ( a ` 0 ) = ( b ` 0 ) -> ( ( a ` 0 ) = ( s ` 0 ) <-> ( b ` 0 ) = ( s ` 0 ) ) )
40	39	rexbidv 3052	. . . . . . . . . . . . . . . 16 |- ( ( a ` 0 ) = ( b ` 0 ) -> ( E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) <-> E. s e. ( `' S " { h } ) ( b ` 0 ) = ( s ` 0 ) ) )
41	40	imbi2d 330	. . . . . . . . . . . . . . 15 |- ( ( a ` 0 ) = ( b ` 0 ) -> ( ( E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) -> E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) ) <-> ( E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) -> E. s e. ( `' S " { h } ) ( b ` 0 ) = ( s ` 0 ) ) ) )
42	38, 41	syl5ibrcom 237	. . . . . . . . . . . . . 14 |- ( ( r e. ( `' S " { g } ) /\ b e. ( `' S " { g } ) ) -> ( ( a ` 0 ) = ( b ` 0 ) -> ( E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) -> E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) ) ) )
43	42	rexlimdva 3031	. . . . . . . . . . . . 13 |- ( r e. ( `' S " { g } ) -> ( E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) -> ( E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) -> E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) ) ) )
44	43	impd 447	. . . . . . . . . . . 12 |- ( r e. ( `' S " { g } ) -> ( ( E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) /\ E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) ) -> E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) ) )
45	44	rexlimiv 3027	. . . . . . . . . . 11 |- ( E. r e. ( `' S " { g } ) ( E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) /\ E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) ) -> E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) )
46	45	reximi 3011	. . . . . . . . . 10 |- ( E. a e. ( `' S " { f } ) E. r e. ( `' S " { g } ) ( E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) /\ E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) ) -> E. a e. ( `' S " { f } ) E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) )
47	21, 46	sylbir 225	. . . . . . . . 9 |- ( ( E. a e. ( `' S " { f } ) E. b e. ( `' S " { g } ) ( a ` 0 ) = ( b ` 0 ) /\ E. r e. ( `' S " { g } ) E. s e. ( `' S " { h } ) ( r ` 0 ) = ( s ` 0 ) ) -> E. a e. ( `' S " { f } ) E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) )
48	16, 20, 47	syl2anb 496	. . . . . . . 8 |- ( ( f L g /\ g L h ) -> E. a e. ( `' S " { f } ) E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) )
49	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18162	. . . . . . . 8 |- ( f L h <-> E. a e. ( `' S " { f } ) E. s e. ( `' S " { h } ) ( a ` 0 ) = ( s ` 0 ) )
50	48, 49	sylibr 224	. . . . . . 7 |- ( ( f L g /\ g L h ) -> f L h )
51	50	adantl 482	. . . . . 6 |- ( ( T. /\ ( f L g /\ g L h ) ) -> f L h )
52		eqid 2622	. . . . . . . . . . . 12 |- ( a ` 0 ) = ( a ` 0 )
53		fveq1 6190	. . . . . . . . . . . . . 14 |- ( b = a -> ( b ` 0 ) = ( a ` 0 ) )
54	53	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( b = a -> ( ( a ` 0 ) = ( b ` 0 ) <-> ( a ` 0 ) = ( a ` 0 ) ) )
55	54	rspcev 3309	. . . . . . . . . . . 12 |- ( ( a e. ( `' S " { f } ) /\ ( a ` 0 ) = ( a ` 0 ) ) -> E. b e. ( `' S " { f } ) ( a ` 0 ) = ( b ` 0 ) )
56	52, 55	mpan2 707	. . . . . . . . . . 11 |- ( a e. ( `' S " { f } ) -> E. b e. ( `' S " { f } ) ( a ` 0 ) = ( b ` 0 ) )
57	56	pm4.71i 664	. . . . . . . . . 10 |- ( a e. ( `' S " { f } ) <-> ( a e. ( `' S " { f } ) /\ E. b e. ( `' S " { f } ) ( a ` 0 ) = ( b ` 0 ) ) )
58		fniniseg 6338	. . . . . . . . . . 11 |- ( S Fn dom S -> ( a e. ( `' S " { f } ) <-> ( a e. dom S /\ ( S ` a ) = f ) ) )
59	24, 58	ax-mp 5	. . . . . . . . . 10 |- ( a e. ( `' S " { f } ) <-> ( a e. dom S /\ ( S ` a ) = f ) )
60	57, 59	bitr3i 266	. . . . . . . . 9 |- ( ( a e. ( `' S " { f } ) /\ E. b e. ( `' S " { f } ) ( a ` 0 ) = ( b ` 0 ) ) <-> ( a e. dom S /\ ( S ` a ) = f ) )
61	60	rexbii2 3039	. . . . . . . 8 |- ( E. a e. ( `' S " { f } ) E. b e. ( `' S " { f } ) ( a ` 0 ) = ( b ` 0 ) <-> E. a e. dom S ( S ` a ) = f )
62	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18162	. . . . . . . 8 |- ( f L f <-> E. a e. ( `' S " { f } ) E. b e. ( `' S " { f } ) ( a ` 0 ) = ( b ` 0 ) )
63		forn 6118	. . . . . . . . . . 11 |- ( S : dom S -onto-> W -> ran S = W )
64	22, 63	ax-mp 5	. . . . . . . . . 10 |- ran S = W
65	64	eleq2i 2693	. . . . . . . . 9 |- ( f e. ran S <-> f e. W )
66		fvelrnb 6243	. . . . . . . . . 10 |- ( S Fn dom S -> ( f e. ran S <-> E. a e. dom S ( S ` a ) = f ) )
67	24, 66	ax-mp 5	. . . . . . . . 9 |- ( f e. ran S <-> E. a e. dom S ( S ` a ) = f )
68	65, 67	bitr3i 266	. . . . . . . 8 |- ( f e. W <-> E. a e. dom S ( S ` a ) = f )
69	61, 62, 68	3bitr4ri 293	. . . . . . 7 |- ( f e. W <-> f L f )
70	69	a1i 11	. . . . . 6 |- ( T. -> ( f e. W <-> f L f ) )
71	8, 19, 51, 70	iserd 7768	. . . . 5 |- ( T. -> L Er W )
72	71	trud 1493	. . . 4 |- L Er W
73		simpl 473	. . . . . . . . . . 11 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> a e. W )
74		foelrn 6378	. . . . . . . . . . 11 |- ( ( S : dom S -onto-> W /\ a e. W ) -> E. r e. dom S a = ( S ` r ) )
75	22, 73, 74	sylancr 695	. . . . . . . . . 10 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> E. r e. dom S a = ( S ` r ) )
76		simprl 794	. . . . . . . . . . . . . 14 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> r e. dom S )
77		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> a = ( S ` r ) )
78	77	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( S ` r ) = a )
79		fniniseg 6338	. . . . . . . . . . . . . . 15 |- ( S Fn dom S -> ( r e. ( `' S " { a } ) <-> ( r e. dom S /\ ( S ` r ) = a ) ) )
80	24, 79	ax-mp 5	. . . . . . . . . . . . . 14 |- ( r e. ( `' S " { a } ) <-> ( r e. dom S /\ ( S ` r ) = a ) )
81	76, 78, 80	sylanbrc 698	. . . . . . . . . . . . 13 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> r e. ( `' S " { a } ) )
82		simplr 792	. . . . . . . . . . . . . . . . 17 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> b e. ran ( T ` a ) )
83	77	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( T ` a ) = ( T ` ( S ` r ) ) )
84	83	rneqd 5353	. . . . . . . . . . . . . . . . 17 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ran ( T ` a ) = ran ( T ` ( S ` r ) ) )
85	82, 84	eleqtrd 2703	. . . . . . . . . . . . . . . 16 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> b e. ran ( T ` ( S ` r ) ) )
86	1, 2, 3, 4, 14, 15	efgsp1 18150	. . . . . . . . . . . . . . . 16 |- ( ( r e. dom S /\ b e. ran ( T ` ( S ` r ) ) ) -> ( r ++ <" b "> ) e. dom S )
87	76, 85, 86	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( r ++ <" b "> ) e. dom S )
88	1, 2, 3, 4, 14, 15	efgsdm 18143	. . . . . . . . . . . . . . . . . . 19 |- ( r e. dom S <-> ( r e. (Word W \ { (/) } ) /\ ( r ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` r ) ) ( r ` i ) e. ran ( T ` ( r ` ( i - 1 ) ) ) ) )
89	88	simp1bi 1076	. . . . . . . . . . . . . . . . . 18 |- ( r e. dom S -> r e. (Word W \ { (/) } ) )
90	89	ad2antrl 764	. . . . . . . . . . . . . . . . 17 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> r e. (Word W \ { (/) } ) )
91	90	eldifad 3586	. . . . . . . . . . . . . . . 16 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> r e. Word W )
92	1, 2, 3, 4	efgtf 18135	. . . . . . . . . . . . . . . . . . . 20 |- ( a e. W -> ( ( T ` a ) = ( f e. ( 0 ... ( # ` a ) ) , g e. ( I X. 2o ) |-> ( a splice <. f , f , <" g ( M ` g ) "> >. ) ) /\ ( T ` a ) : ( ( 0 ... ( # ` a ) ) X. ( I X. 2o ) ) --> W ) )
93	92	simprd 479	. . . . . . . . . . . . . . . . . . 19 |- ( a e. W -> ( T ` a ) : ( ( 0 ... ( # ` a ) ) X. ( I X. 2o ) ) --> W )
94		frn 6053	. . . . . . . . . . . . . . . . . . 19 |- ( ( T ` a ) : ( ( 0 ... ( # ` a ) ) X. ( I X. 2o ) ) --> W -> ran ( T ` a ) C_ W )
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( a e. W -> ran ( T ` a ) C_ W )
96	95	sselda 3603	. . . . . . . . . . . . . . . . 17 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> b e. W )
97	96	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> b e. W )
98	1, 2, 3, 4, 14, 15	efgsval2 18146	. . . . . . . . . . . . . . . 16 |- ( ( r e. Word W /\ b e. W /\ ( r ++ <" b "> ) e. dom S ) -> ( S ` ( r ++ <" b "> ) ) = b )
99	91, 97, 87, 98	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( S ` ( r ++ <" b "> ) ) = b )
100		fniniseg 6338	. . . . . . . . . . . . . . . 16 |- ( S Fn dom S -> ( ( r ++ <" b "> ) e. ( `' S " { b } ) <-> ( ( r ++ <" b "> ) e. dom S /\ ( S ` ( r ++ <" b "> ) ) = b ) ) )
101	24, 100	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( ( r ++ <" b "> ) e. ( `' S " { b } ) <-> ( ( r ++ <" b "> ) e. dom S /\ ( S ` ( r ++ <" b "> ) ) = b ) )
102	87, 99, 101	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( r ++ <" b "> ) e. ( `' S " { b } ) )
103	97	s1cld 13383	. . . . . . . . . . . . . . . 16 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> <" b "> e. Word W )
104		eldifsn 4317	. . . . . . . . . . . . . . . . . . 19 |- ( r e. (Word W \ { (/) } ) <-> ( r e. Word W /\ r =/= (/) ) )
105		lennncl 13325	. . . . . . . . . . . . . . . . . . 19 |- ( ( r e. Word W /\ r =/= (/) ) -> ( # ` r ) e. NN )
106	104, 105	sylbi 207	. . . . . . . . . . . . . . . . . 18 |- ( r e. (Word W \ { (/) } ) -> ( # ` r ) e. NN )
107	90, 106	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( # ` r ) e. NN )
108		lbfzo0 12507	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ( 0..^ ( # ` r ) ) <-> ( # ` r ) e. NN )
109	107, 108	sylibr 224	. . . . . . . . . . . . . . . 16 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> 0 e. ( 0..^ ( # ` r ) ) )
110		ccatval1 13361	. . . . . . . . . . . . . . . 16 |- ( ( r e. Word W /\ <" b "> e. Word W /\ 0 e. ( 0..^ ( # ` r ) ) ) -> ( ( r ++ <" b "> ) ` 0 ) = ( r ` 0 ) )
111	91, 103, 109, 110	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( ( r ++ <" b "> ) ` 0 ) = ( r ` 0 ) )
112	111	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( r ` 0 ) = ( ( r ++ <" b "> ) ` 0 ) )
113		fveq1 6190	. . . . . . . . . . . . . . . 16 |- ( s = ( r ++ <" b "> ) -> ( s ` 0 ) = ( ( r ++ <" b "> ) ` 0 ) )
114	113	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( s = ( r ++ <" b "> ) -> ( ( r ` 0 ) = ( s ` 0 ) <-> ( r ` 0 ) = ( ( r ++ <" b "> ) ` 0 ) ) )
115	114	rspcev 3309	. . . . . . . . . . . . . 14 |- ( ( ( r ++ <" b "> ) e. ( `' S " { b } ) /\ ( r ` 0 ) = ( ( r ++ <" b "> ) ` 0 ) ) -> E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) )
116	102, 112, 115	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) )
117	81, 116	jca 554	. . . . . . . . . . . 12 |- ( ( ( a e. W /\ b e. ran ( T ` a ) ) /\ ( r e. dom S /\ a = ( S ` r ) ) ) -> ( r e. ( `' S " { a } ) /\ E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) ) )
118	117	ex 450	. . . . . . . . . . 11 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> ( ( r e. dom S /\ a = ( S ` r ) ) -> ( r e. ( `' S " { a } ) /\ E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) ) ) )
119	118	reximdv2 3014	. . . . . . . . . 10 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> ( E. r e. dom S a = ( S ` r ) -> E. r e. ( `' S " { a } ) E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) ) )
120	75, 119	mpd 15	. . . . . . . . 9 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> E. r e. ( `' S " { a } ) E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) )
121	1, 2, 3, 4, 14, 15, 6	efgrelexlema 18162	. . . . . . . . 9 |- ( a L b <-> E. r e. ( `' S " { a } ) E. s e. ( `' S " { b } ) ( r ` 0 ) = ( s ` 0 ) )
122	120, 121	sylibr 224	. . . . . . . 8 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> a L b )
123		vex 3203	. . . . . . . . 9 |- b e. _V
124		vex 3203	. . . . . . . . 9 |- a e. _V
125	123, 124	elec 7786	. . . . . . . 8 |- ( b e. [ a ] L <-> a L b )
126	122, 125	sylibr 224	. . . . . . 7 |- ( ( a e. W /\ b e. ran ( T ` a ) ) -> b e. [ a ] L )
127	126	ex 450	. . . . . 6 |- ( a e. W -> ( b e. ran ( T ` a ) -> b e. [ a ] L ) )
128	127	ssrdv 3609	. . . . 5 |- ( a e. W -> ran ( T ` a ) C_ [ a ] L )
129	128	rgen 2922	. . . 4 |- A. a e. W ran ( T ` a ) C_ [ a ] L
130		fvex 6201	. . . . . . 7 |- ( _I ` Word ( I X. 2o ) ) e. _V
131	1, 130	eqeltri 2697	. . . . . 6 |- W e. _V
132		erex 7766	. . . . . 6 |- ( L Er W -> ( W e. _V -> L e. _V ) )
133	72, 131, 132	mp2 9	. . . . 5 |- L e. _V
134		ereq1 7749	. . . . . 6 |- ( r = L -> ( r Er W <-> L Er W ) )
135		eceq2 7784	. . . . . . . 8 |- ( r = L -> [ a ] r = [ a ] L )
136	135	sseq2d 3633	. . . . . . 7 |- ( r = L -> ( ran ( T ` a ) C_ [ a ] r <-> ran ( T ` a ) C_ [ a ] L ) )
137	136	ralbidv 2986	. . . . . 6 |- ( r = L -> ( A. a e. W ran ( T ` a ) C_ [ a ] r <-> A. a e. W ran ( T ` a ) C_ [ a ] L ) )
138	134, 137	anbi12d 747	. . . . 5 |- ( r = L -> ( ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) <-> ( L Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] L ) ) )
139	133, 138	elab 3350	. . . 4 |- ( L e. { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) } <-> ( L Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] L ) )
140	72, 129, 139	mpbir2an 955	. . 3 |- L e. { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) }
141		intss1 4492	. . 3 |- ( L e. { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) } -> |^| { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) } C_ L )
142	140, 141	ax-mp 5	. 2 |- |^| { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) } C_ L
143	5, 142	eqsstri 3635	1 |- .~ C_ L

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   |^|cint 4475   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Rel wrel 5119   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   [cec 7740   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   splice csplice 13296   <"cs2 13586   ~_FG cefg 18119

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-efg 18122

This theorem is referenced by:  efgrelex  18164