Theorem efgrelexlema 18162

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
efgrelexlema	|- ( A L B <-> E. a e. ( `' S " { A } ) E. b e. ( `' S " { B } ) ( a ` 0 ) = ( b ` 0 ) )

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

Allowed substitution hints:   A( x, y, z, w, v, t, k, m, n)   B( x, y, z, w, v, t, k, m, n)   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 efgrelexlema

Step	Hyp	Ref	Expression
1		efgrelexlem.1	. . 3 |- L = { <. i , j >. | E. c e. ( `' S " { i } ) E. d e. ( `' S " { j } ) ( c ` 0 ) = ( d ` 0 ) }
2	1	bropaex12 5192	. 2 |- ( A L B -> ( A e. _V /\ B e. _V ) )
3		n0i 3920	. . . . . 6 |- ( a e. ( `' S " { A } ) -> -. ( `' S " { A } ) = (/) )
4		snprc 4253	. . . . . . . 8 |- ( -. A e. _V <-> { A } = (/) )
5		imaeq2 5462	. . . . . . . 8 |- ( { A } = (/) -> ( `' S " { A } ) = ( `' S " (/) ) )
6	4, 5	sylbi 207	. . . . . . 7 |- ( -. A e. _V -> ( `' S " { A } ) = ( `' S " (/) ) )
7		ima0 5481	. . . . . . 7 |- ( `' S " (/) ) = (/)
8	6, 7	syl6eq 2672	. . . . . 6 |- ( -. A e. _V -> ( `' S " { A } ) = (/) )
9	3, 8	nsyl2 142	. . . . 5 |- ( a e. ( `' S " { A } ) -> A e. _V )
10		n0i 3920	. . . . . 6 |- ( b e. ( `' S " { B } ) -> -. ( `' S " { B } ) = (/) )
11		snprc 4253	. . . . . . . 8 |- ( -. B e. _V <-> { B } = (/) )
12		imaeq2 5462	. . . . . . . 8 |- ( { B } = (/) -> ( `' S " { B } ) = ( `' S " (/) ) )
13	11, 12	sylbi 207	. . . . . . 7 |- ( -. B e. _V -> ( `' S " { B } ) = ( `' S " (/) ) )
14	13, 7	syl6eq 2672	. . . . . 6 |- ( -. B e. _V -> ( `' S " { B } ) = (/) )
15	10, 14	nsyl2 142	. . . . 5 |- ( b e. ( `' S " { B } ) -> B e. _V )
16	9, 15	anim12i 590	. . . 4 |- ( ( a e. ( `' S " { A } ) /\ b e. ( `' S " { B } ) ) -> ( A e. _V /\ B e. _V ) )
17	16	a1d 25	. . 3 |- ( ( a e. ( `' S " { A } ) /\ b e. ( `' S " { B } ) ) -> ( ( a ` 0 ) = ( b ` 0 ) -> ( A e. _V /\ B e. _V ) ) )
18	17	rexlimivv 3036	. 2 |- ( E. a e. ( `' S " { A } ) E. b e. ( `' S " { B } ) ( a ` 0 ) = ( b ` 0 ) -> ( A e. _V /\ B e. _V ) )
19		fveq1 6190	. . . . . 6 |- ( c = a -> ( c ` 0 ) = ( a ` 0 ) )
20	19	eqeq1d 2624	. . . . 5 |- ( c = a -> ( ( c ` 0 ) = ( d ` 0 ) <-> ( a ` 0 ) = ( d ` 0 ) ) )
21		fveq1 6190	. . . . . 6 |- ( d = b -> ( d ` 0 ) = ( b ` 0 ) )
22	21	eqeq2d 2632	. . . . 5 |- ( d = b -> ( ( a ` 0 ) = ( d ` 0 ) <-> ( a ` 0 ) = ( b ` 0 ) ) )
23	20, 22	cbvrex2v 3180	. . . 4 |- ( E. c e. ( `' S " { i } ) E. d e. ( `' S " { j } ) ( c ` 0 ) = ( d ` 0 ) <-> E. a e. ( `' S " { i } ) E. b e. ( `' S " { j } ) ( a ` 0 ) = ( b ` 0 ) )
24		sneq 4187	. . . . . 6 |- ( i = A -> { i } = { A } )
25	24	imaeq2d 5466	. . . . 5 |- ( i = A -> ( `' S " { i } ) = ( `' S " { A } ) )
26	25	rexeqdv 3145	. . . 4 |- ( i = A -> ( E. a e. ( `' S " { i } ) E. b e. ( `' S " { j } ) ( a ` 0 ) = ( b ` 0 ) <-> E. a e. ( `' S " { A } ) E. b e. ( `' S " { j } ) ( a ` 0 ) = ( b ` 0 ) ) )
27	23, 26	syl5bb 272	. . 3 |- ( i = A -> ( E. c e. ( `' S " { i } ) E. d e. ( `' S " { j } ) ( c ` 0 ) = ( d ` 0 ) <-> E. a e. ( `' S " { A } ) E. b e. ( `' S " { j } ) ( a ` 0 ) = ( b ` 0 ) ) )
28		sneq 4187	. . . . . 6 |- ( j = B -> { j } = { B } )
29	28	imaeq2d 5466	. . . . 5 |- ( j = B -> ( `' S " { j } ) = ( `' S " { B } ) )
30	29	rexeqdv 3145	. . . 4 |- ( j = B -> ( E. b e. ( `' S " { j } ) ( a ` 0 ) = ( b ` 0 ) <-> E. b e. ( `' S " { B } ) ( a ` 0 ) = ( b ` 0 ) ) )
31	30	rexbidv 3052	. . 3 |- ( j = B -> ( E. a e. ( `' S " { A } ) E. b e. ( `' S " { j } ) ( a ` 0 ) = ( b ` 0 ) <-> E. a e. ( `' S " { A } ) E. b e. ( `' S " { B } ) ( a ` 0 ) = ( b ` 0 ) ) )
32	27, 31, 1	brabg 4994	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A L B <-> E. a e. ( `' S " { A } ) E. b e. ( `' S " { B } ) ( a ` 0 ) = ( b ` 0 ) ) )
33	2, 18, 32	pm5.21nii 368	1 |- ( A L B <-> E. a e. ( `' S " { A } ) E. b e. ( `' S " { B } ) ( a ` 0 ) = ( b ` 0 ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   1c1 9937   - cmin 10266   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by:  efgrelexlemb  18163  efgrelex  18164