Theorem efginvrel1 18141

Description: The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (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 ) "> >. ) ) )

Assertion

Ref	Expression
efginvrel1	|- ( A e. W -> ( ( M o. (reverse ` A ) ) ++ A ) .~ (/) )

Distinct variable groups:   y, z   v, n, w, y, z   n, M, v, w   n, W, v, w, y, z   y, .~ , z   n, I, v, w, y, z

Allowed substitution hints:   A( y, z, w, v, n)   .~ ( w, v, n)   T( y, z, w, v, n)   M( y, z)

Proof of Theorem efginvrel1

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

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . . . 10 |- W = ( _I ` Word ( I X. 2o ) )
2		fviss 6256	. . . . . . . . . 10 |- ( _I ` Word ( I X. 2o ) ) C_ Word ( I X. 2o )
3	1, 2	eqsstri 3635	. . . . . . . . 9 |- W C_ Word ( I X. 2o )
4	3	sseli 3599	. . . . . . . 8 |- ( A e. W -> A e. Word ( I X. 2o ) )
5		revcl 13510	. . . . . . . 8 |- ( A e. Word ( I X. 2o ) -> (reverse ` A ) e. Word ( I X. 2o ) )
6	4, 5	syl 17	. . . . . . 7 |- ( A e. W -> (reverse ` A ) e. Word ( I X. 2o ) )
7		efgval2.m	. . . . . . . 8 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
8	7	efgmf 18126	. . . . . . 7 |- M : ( I X. 2o ) --> ( I X. 2o )
9		revco 13580	. . . . . . 7 |- ( ( (reverse ` A ) e. Word ( I X. 2o ) /\ M : ( I X. 2o ) --> ( I X. 2o ) ) -> ( M o. (reverse ` (reverse ` A ) ) ) = (reverse ` ( M o. (reverse ` A ) ) ) )
10	6, 8, 9	sylancl 694	. . . . . 6 |- ( A e. W -> ( M o. (reverse ` (reverse ` A ) ) ) = (reverse ` ( M o. (reverse ` A ) ) ) )
11		revrev 13516	. . . . . . . 8 |- ( A e. Word ( I X. 2o ) -> (reverse ` (reverse ` A ) ) = A )
12	4, 11	syl 17	. . . . . . 7 |- ( A e. W -> (reverse ` (reverse ` A ) ) = A )
13	12	coeq2d 5284	. . . . . 6 |- ( A e. W -> ( M o. (reverse ` (reverse ` A ) ) ) = ( M o. A ) )
14	10, 13	eqtr3d 2658	. . . . 5 |- ( A e. W -> (reverse ` ( M o. (reverse ` A ) ) ) = ( M o. A ) )
15	14	coeq2d 5284	. . . 4 |- ( A e. W -> ( M o. (reverse ` ( M o. (reverse ` A ) ) ) ) = ( M o. ( M o. A ) ) )
16		wrdf 13310	. . . . . . . . 9 |- ( A e. Word ( I X. 2o ) -> A : ( 0..^ ( # ` A ) ) --> ( I X. 2o ) )
17	4, 16	syl 17	. . . . . . . 8 |- ( A e. W -> A : ( 0..^ ( # ` A ) ) --> ( I X. 2o ) )
18	17	ffvelrnda 6359	. . . . . . 7 |- ( ( A e. W /\ c e. ( 0..^ ( # ` A ) ) ) -> ( A ` c ) e. ( I X. 2o ) )
19	7	efgmnvl 18127	. . . . . . 7 |- ( ( A ` c ) e. ( I X. 2o ) -> ( M ` ( M ` ( A ` c ) ) ) = ( A ` c ) )
20	18, 19	syl 17	. . . . . 6 |- ( ( A e. W /\ c e. ( 0..^ ( # ` A ) ) ) -> ( M ` ( M ` ( A ` c ) ) ) = ( A ` c ) )
21	20	mpteq2dva 4744	. . . . 5 |- ( A e. W -> ( c e. ( 0..^ ( # ` A ) ) |-> ( M ` ( M ` ( A ` c ) ) ) ) = ( c e. ( 0..^ ( # ` A ) ) |-> ( A ` c ) ) )
22	8	ffvelrni 6358	. . . . . . 7 |- ( ( A ` c ) e. ( I X. 2o ) -> ( M ` ( A ` c ) ) e. ( I X. 2o ) )
23	18, 22	syl 17	. . . . . 6 |- ( ( A e. W /\ c e. ( 0..^ ( # ` A ) ) ) -> ( M ` ( A ` c ) ) e. ( I X. 2o ) )
24		fcompt 6400	. . . . . . 7 |- ( ( M : ( I X. 2o ) --> ( I X. 2o ) /\ A : ( 0..^ ( # ` A ) ) --> ( I X. 2o ) ) -> ( M o. A ) = ( c e. ( 0..^ ( # ` A ) ) |-> ( M ` ( A ` c ) ) ) )
25	8, 17, 24	sylancr 695	. . . . . 6 |- ( A e. W -> ( M o. A ) = ( c e. ( 0..^ ( # ` A ) ) |-> ( M ` ( A ` c ) ) ) )
26	8	a1i 11	. . . . . . 7 |- ( A e. W -> M : ( I X. 2o ) --> ( I X. 2o ) )
27	26	feqmptd 6249	. . . . . 6 |- ( A e. W -> M = ( a e. ( I X. 2o ) |-> ( M ` a ) ) )
28		fveq2 6191	. . . . . 6 |- ( a = ( M ` ( A ` c ) ) -> ( M ` a ) = ( M ` ( M ` ( A ` c ) ) ) )
29	23, 25, 27, 28	fmptco 6396	. . . . 5 |- ( A e. W -> ( M o. ( M o. A ) ) = ( c e. ( 0..^ ( # ` A ) ) |-> ( M ` ( M ` ( A ` c ) ) ) ) )
30	17	feqmptd 6249	. . . . 5 |- ( A e. W -> A = ( c e. ( 0..^ ( # ` A ) ) |-> ( A ` c ) ) )
31	21, 29, 30	3eqtr4d 2666	. . . 4 |- ( A e. W -> ( M o. ( M o. A ) ) = A )
32	15, 31	eqtrd 2656	. . 3 |- ( A e. W -> ( M o. (reverse ` ( M o. (reverse ` A ) ) ) ) = A )
33	32	oveq2d 6666	. 2 |- ( A e. W -> ( ( M o. (reverse ` A ) ) ++ ( M o. (reverse ` ( M o. (reverse ` A ) ) ) ) ) = ( ( M o. (reverse ` A ) ) ++ A ) )
34		wrdco 13577	. . . . 5 |- ( ( (reverse ` A ) e. Word ( I X. 2o ) /\ M : ( I X. 2o ) --> ( I X. 2o ) ) -> ( M o. (reverse ` A ) ) e. Word ( I X. 2o ) )
35	6, 8, 34	sylancl 694	. . . 4 |- ( A e. W -> ( M o. (reverse ` A ) ) e. Word ( I X. 2o ) )
36	1	efgrcl 18128	. . . . 5 |- ( A e. W -> ( I e. _V /\ W = Word ( I X. 2o ) ) )
37	36	simprd 479	. . . 4 |- ( A e. W -> W = Word ( I X. 2o ) )
38	35, 37	eleqtrrd 2704	. . 3 |- ( A e. W -> ( M o. (reverse ` A ) ) e. W )
39		efgval.r	. . . 4 |- .~ = ( ~FG ` I )
40		efgval2.t	. . . 4 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
41	1, 39, 7, 40	efginvrel2 18140	. . 3 |- ( ( M o. (reverse ` A ) ) e. W -> ( ( M o. (reverse ` A ) ) ++ ( M o. (reverse ` ( M o. (reverse ` A ) ) ) ) ) .~ (/) )
42	38, 41	syl 17	. 2 |- ( A e. W -> ( ( M o. (reverse ` A ) ) ++ ( M o. (reverse ` ( M o. (reverse ` A ) ) ) ) ) .~ (/) )
43	33, 42	eqbrtrrd 4677	1 |- ( A e. W -> ( ( M o. (reverse ` A ) ) ++ A ) .~ (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   <.cop 4183   <.cotp 4185   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   splice csplice 13296  reversecreverse 13297   <"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-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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-efg 18122

This theorem is referenced by:  frgp0  18173