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Theorem frgpinv 15434
Description: The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
frgpadd.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpadd.g  |-  G  =  (freeGrp `  I )
frgpadd.r  |-  .~  =  ( ~FG  `  I )
frgpinv.n  |-  N  =  ( inv g `  G )
frgpinv.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
Assertion
Ref Expression
frgpinv  |-  ( A  e.  W  ->  ( N `  [ A ]  .~  )  =  [
( M  o.  (reverse `  A ) ) ]  .~  )
Distinct variable groups:    y, z, I    y,  .~ , z    y, W, z
Allowed substitution hints:    A( y, z)    G( y, z)    M( y, z)    N( y, z)

Proof of Theorem frgpinv
Dummy variables  n  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpadd.w . . . . . . . . 9  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 fviss 5820 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3367 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
43sseli 3333 . . . . . . 7  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 11831 . . . . . . 7  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  A
)  e. Word  ( I  X.  2o ) )
64, 5syl 16 . . . . . 6  |-  ( A  e.  W  ->  (reverse `  A )  e. Word  (
I  X.  2o ) )
7 frgpinv.m . . . . . . 7  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
87efgmf 15383 . . . . . 6  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 wrdco 11838 . . . . . 6  |-  ( ( (reverse `  A )  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
106, 8, 9sylancl 645 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
111efgrcl 15385 . . . . . 6  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1211simprd 451 . . . . 5  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
1310, 12eleqtrrd 2520 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e.  W
)
14 frgpadd.g . . . . 5  |-  G  =  (freeGrp `  I )
15 frgpadd.r . . . . 5  |-  .~  =  ( ~FG  `  I )
16 eqid 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
171, 14, 15, 16frgpadd 15433 . . . 4  |-  ( ( A  e.  W  /\  ( M  o.  (reverse `  A ) )  e.  W )  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  [
( A concat  ( M  o.  (reverse `  A )
) ) ]  .~  )
1813, 17mpdan 651 . . 3  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  [
( A concat  ( M  o.  (reverse `  A )
) ) ]  .~  )
191, 15efger 15388 . . . . 5  |-  .~  Er  W
2019a1i 11 . . . 4  |-  ( A  e.  W  ->  .~  Er  W )
21 eqid 2443 . . . . 5  |-  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >.
) ) )  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
221, 15, 7, 21efginvrel2 15397 . . . 4  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
2320, 22erthi 6987 . . 3  |-  ( A  e.  W  ->  [ ( A concat  ( M  o.  (reverse `  A ) ) ) ]  .~  =  [ (/) ]  .~  )
2414, 15frgp0 15430 . . . . . 6  |-  ( I  e.  _V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2524adantr 453 . . . . 5  |-  ( ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) )  -> 
( G  e.  Grp  /\ 
[ (/) ]  .~  =  ( 0g `  G ) ) )
2611, 25syl 16 . . . 4  |-  ( A  e.  W  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2726simprd 451 . . 3  |-  ( A  e.  W  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
2818, 23, 273eqtrd 2479 . 2  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) )
2926simpld 447 . . 3  |-  ( A  e.  W  ->  G  e.  Grp )
30 eqid 2443 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
3114, 15, 1, 30frgpeccl 15431 . . 3  |-  ( A  e.  W  ->  [ A ]  .~  e.  ( Base `  G ) )
3214, 15, 1, 30frgpeccl 15431 . . . 4  |-  ( ( M  o.  (reverse `  A
) )  e.  W  ->  [ ( M  o.  (reverse `  A ) ) ]  .~  e.  (
Base `  G )
)
3313, 32syl 16 . . 3  |-  ( A  e.  W  ->  [ ( M  o.  (reverse `  A
) ) ]  .~  e.  ( Base `  G
) )
34 eqid 2443 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
35 frgpinv.n . . . 4  |-  N  =  ( inv g `  G )
3630, 16, 34, 35grpinvid1 14891 . . 3  |-  ( ( G  e.  Grp  /\  [ A ]  .~  e.  ( Base `  G )  /\  [ ( M  o.  (reverse `  A ) ) ]  .~  e.  (
Base `  G )
)  ->  ( ( N `  [ A ]  .~  )  =  [
( M  o.  (reverse `  A ) ) ]  .~  <->  ( [ A ]  .~  ( +g  `  G
) [ ( M  o.  (reverse `  A
) ) ]  .~  )  =  ( 0g `  G ) ) )
3729, 31, 33, 36syl3anc 1185 . 2  |-  ( A  e.  W  ->  (
( N `  [ A ]  .~  )  =  [ ( M  o.  (reverse `  A ) ) ]  .~  <->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) ) )
3828, 37mpbird 225 1  |-  ( A  e.  W  ->  ( N `  [ A ]  .~  )  =  [
( M  o.  (reverse `  A ) ) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728   _Vcvv 2965    \ cdif 3306   (/)c0 3616   <.cop 3846   <.cotp 3847    e. cmpt 4297    _I cid 4528    X. cxp 4911    o. ccom 4917   -->wf 5485   ` cfv 5489  (class class class)co 6117    e. cmpt2 6119   1oc1o 6753   2oc2o 6754    Er wer 6938   [cec 6939   0cc0 9028   ...cfz 11081   #chash 11656  Word cword 11755   concat cconcat 11756   splice csplice 11759  reversecreverse 11760   <"cs2 11843   Basecbs 13507   +g cplusg 13567   0gc0g 13761   Grpcgrp 14723   inv gcminusg 14724   ~FG cefg 15376  freeGrpcfrgp 15377
This theorem is referenced by:  vrgpinv  15439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-ot 3853  df-uni 4045  df-int 4080  df-iun 4124  df-iin 4125  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-2o 6761  df-oadd 6764  df-er 6941  df-ec 6943  df-qs 6947  df-map 7056  df-pm 7057  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-sup 7482  df-card 7864  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-3 10097  df-4 10098  df-5 10099  df-6 10100  df-7 10101  df-8 10102  df-9 10103  df-10 10104  df-n0 10260  df-z 10321  df-dec 10421  df-uz 10527  df-fz 11082  df-fzo 11174  df-hash 11657  df-word 11761  df-concat 11762  df-s1 11763  df-substr 11764  df-splice 11765  df-reverse 11766  df-s2 11850  df-struct 13509  df-ndx 13510  df-slot 13511  df-base 13512  df-plusg 13580  df-mulr 13581  df-sca 13583  df-vsca 13584  df-tset 13586  df-ple 13587  df-ds 13589  df-0g 13765  df-imas 13772  df-divs 13773  df-mnd 14728  df-frmd 14832  df-grp 14850  df-minusg 14851  df-efg 15379  df-frgp 15380
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