Theorem efgredeu 18165

Description: There is a unique reduced word equivalent to a given word. (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 ) ) )

Assertion

Ref	Expression
efgredeu	|- ( A e. W -> E! d e. D d .~ A )

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

Allowed substitution hints:   A( 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)   M( y, z, k, d)

Proof of Theorem efgredeu

Dummy variables a b c i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . 5 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . . 5 |- .~ = ( ~FG ` I )
3		efgval2.m	. . . . 5 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . . . 5 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5		efgred.d	. . . . 5 |- D = ( W \ U_ x e. W ran ( T ` x ) )
6		efgred.s	. . . . 5 |- 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 ) ) )
7	1, 2, 3, 4, 5, 6	efgsfo 18152	. . . 4 |- S : dom S -onto-> W
8		foelrn 6378	. . . 4 |- ( ( S : dom S -onto-> W /\ A e. W ) -> E. a e. dom S A = ( S ` a ) )
9	7, 8	mpan 706	. . 3 |- ( A e. W -> E. a e. dom S A = ( S ` a ) )
10	1, 2, 3, 4, 5, 6	efgsdm 18143	. . . . . . . 8 |- ( a e. dom S <-> ( a e. (Word W \ { (/) } ) /\ ( a ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` a ) ) ( a ` i ) e. ran ( T ` ( a ` ( i - 1 ) ) ) ) )
11	10	simp2bi 1077	. . . . . . 7 |- ( a e. dom S -> ( a ` 0 ) e. D )
12	11	adantl 482	. . . . . 6 |- ( ( A e. W /\ a e. dom S ) -> ( a ` 0 ) e. D )
13	1, 2, 3, 4, 5, 6	efgsrel 18147	. . . . . . 7 |- ( a e. dom S -> ( a ` 0 ) .~ ( S ` a ) )
14	13	adantl 482	. . . . . 6 |- ( ( A e. W /\ a e. dom S ) -> ( a ` 0 ) .~ ( S ` a ) )
15		breq1 4656	. . . . . . 7 |- ( d = ( a ` 0 ) -> ( d .~ ( S ` a ) <-> ( a ` 0 ) .~ ( S ` a ) ) )
16	15	rspcev 3309	. . . . . 6 |- ( ( ( a ` 0 ) e. D /\ ( a ` 0 ) .~ ( S ` a ) ) -> E. d e. D d .~ ( S ` a ) )
17	12, 14, 16	syl2anc 693	. . . . 5 |- ( ( A e. W /\ a e. dom S ) -> E. d e. D d .~ ( S ` a ) )
18		breq2 4657	. . . . . 6 |- ( A = ( S ` a ) -> ( d .~ A <-> d .~ ( S ` a ) ) )
19	18	rexbidv 3052	. . . . 5 |- ( A = ( S ` a ) -> ( E. d e. D d .~ A <-> E. d e. D d .~ ( S ` a ) ) )
20	17, 19	syl5ibrcom 237	. . . 4 |- ( ( A e. W /\ a e. dom S ) -> ( A = ( S ` a ) -> E. d e. D d .~ A ) )
21	20	rexlimdva 3031	. . 3 |- ( A e. W -> ( E. a e. dom S A = ( S ` a ) -> E. d e. D d .~ A ) )
22	9, 21	mpd 15	. 2 |- ( A e. W -> E. d e. D d .~ A )
23	1, 2	efger 18131	. . . . . . 7 |- .~ Er W
24	23	a1i 11	. . . . . 6 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> .~ Er W )
25		simprl 794	. . . . . 6 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> d .~ A )
26		simprr 796	. . . . . 6 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> c .~ A )
27	24, 25, 26	ertr4d 7761	. . . . 5 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> d .~ c )
28	1, 2, 3, 4, 5, 6	efgrelex 18164	. . . . . 6 |- ( d .~ c -> E. a e. ( `' S " { d } ) E. b e. ( `' S " { c } ) ( a ` 0 ) = ( b ` 0 ) )
29		fofn 6117	. . . . . . . . . . . . . 14 |- ( S : dom S -onto-> W -> S Fn dom S )
30		fniniseg 6338	. . . . . . . . . . . . . 14 |- ( S Fn dom S -> ( a e. ( `' S " { d } ) <-> ( a e. dom S /\ ( S ` a ) = d ) ) )
31	7, 29, 30	mp2b 10	. . . . . . . . . . . . 13 |- ( a e. ( `' S " { d } ) <-> ( a e. dom S /\ ( S ` a ) = d ) )
32	31	simplbi 476	. . . . . . . . . . . 12 |- ( a e. ( `' S " { d } ) -> a e. dom S )
33	32	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> a e. dom S )
34	1, 2, 3, 4, 5, 6	efgsval 18144	. . . . . . . . . . 11 |- ( a e. dom S -> ( S ` a ) = ( a ` ( ( # ` a ) - 1 ) ) )
35	33, 34	syl 17	. . . . . . . . . 10 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( S ` a ) = ( a ` ( ( # ` a ) - 1 ) ) )
36	31	simprbi 480	. . . . . . . . . . 11 |- ( a e. ( `' S " { d } ) -> ( S ` a ) = d )
37	36	ad2antrl 764	. . . . . . . . . 10 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( S ` a ) = d )
38		simpllr 799	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( d e. D /\ c e. D ) )
39	38	simpld 475	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> d e. D )
40	37, 39	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( S ` a ) e. D )
41	1, 2, 3, 4, 5, 6	efgs1b 18149	. . . . . . . . . . . . . . 15 |- ( a e. dom S -> ( ( S ` a ) e. D <-> ( # ` a ) = 1 ) )
42	33, 41	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( S ` a ) e. D <-> ( # ` a ) = 1 ) )
43	40, 42	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( # ` a ) = 1 )
44	43	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( # ` a ) - 1 ) = ( 1 - 1 ) )
45		1m1e0 11089	. . . . . . . . . . . 12 |- ( 1 - 1 ) = 0
46	44, 45	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( # ` a ) - 1 ) = 0 )
47	46	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( a ` ( ( # ` a ) - 1 ) ) = ( a ` 0 ) )
48	35, 37, 47	3eqtr3rd 2665	. . . . . . . . 9 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( a ` 0 ) = d )
49		fniniseg 6338	. . . . . . . . . . . . . 14 |- ( S Fn dom S -> ( b e. ( `' S " { c } ) <-> ( b e. dom S /\ ( S ` b ) = c ) ) )
50	7, 29, 49	mp2b 10	. . . . . . . . . . . . 13 |- ( b e. ( `' S " { c } ) <-> ( b e. dom S /\ ( S ` b ) = c ) )
51	50	simplbi 476	. . . . . . . . . . . 12 |- ( b e. ( `' S " { c } ) -> b e. dom S )
52	51	ad2antll 765	. . . . . . . . . . 11 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> b e. dom S )
53	1, 2, 3, 4, 5, 6	efgsval 18144	. . . . . . . . . . 11 |- ( b e. dom S -> ( S ` b ) = ( b ` ( ( # ` b ) - 1 ) ) )
54	52, 53	syl 17	. . . . . . . . . 10 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( S ` b ) = ( b ` ( ( # ` b ) - 1 ) ) )
55	50	simprbi 480	. . . . . . . . . . 11 |- ( b e. ( `' S " { c } ) -> ( S ` b ) = c )
56	55	ad2antll 765	. . . . . . . . . 10 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( S ` b ) = c )
57	38	simprd 479	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> c e. D )
58	56, 57	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( S ` b ) e. D )
59	1, 2, 3, 4, 5, 6	efgs1b 18149	. . . . . . . . . . . . . . 15 |- ( b e. dom S -> ( ( S ` b ) e. D <-> ( # ` b ) = 1 ) )
60	52, 59	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( S ` b ) e. D <-> ( # ` b ) = 1 ) )
61	58, 60	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( # ` b ) = 1 )
62	61	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( # ` b ) - 1 ) = ( 1 - 1 ) )
63	62, 45	syl6eq 2672	. . . . . . . . . . 11 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( # ` b ) - 1 ) = 0 )
64	63	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( b ` ( ( # ` b ) - 1 ) ) = ( b ` 0 ) )
65	54, 56, 64	3eqtr3rd 2665	. . . . . . . . 9 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( b ` 0 ) = c )
66	48, 65	eqeq12d 2637	. . . . . . . 8 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( a ` 0 ) = ( b ` 0 ) <-> d = c ) )
67	66	biimpd 219	. . . . . . 7 |- ( ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) /\ ( a e. ( `' S " { d } ) /\ b e. ( `' S " { c } ) ) ) -> ( ( a ` 0 ) = ( b ` 0 ) -> d = c ) )
68	67	rexlimdvva 3038	. . . . . 6 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> ( E. a e. ( `' S " { d } ) E. b e. ( `' S " { c } ) ( a ` 0 ) = ( b ` 0 ) -> d = c ) )
69	28, 68	syl5 34	. . . . 5 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> ( d .~ c -> d = c ) )
70	27, 69	mpd 15	. . . 4 |- ( ( ( A e. W /\ ( d e. D /\ c e. D ) ) /\ ( d .~ A /\ c .~ A ) ) -> d = c )
71	70	ex 450	. . 3 |- ( ( A e. W /\ ( d e. D /\ c e. D ) ) -> ( ( d .~ A /\ c .~ A ) -> d = c ) )
72	71	ralrimivva 2971	. 2 |- ( A e. W -> A. d e. D A. c e. D ( ( d .~ A /\ c .~ A ) -> d = c ) )
73		breq1 4656	. . 3 |- ( d = c -> ( d .~ A <-> c .~ A ) )
74	73	reu4 3400	. 2 |- ( E! d e. D d .~ A <-> ( E. d e. D d .~ A /\ A. d e. D A. c e. D ( ( d .~ A /\ c .~ A ) -> d = c ) ) )
75	22, 72, 74	sylanbrc 698	1 |- ( A e. W -> E! d e. D d .~ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   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-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-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-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-iin 4523  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:  efgred2  18166  frgpnabllem2  18277