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Theorem frgpinv 16254
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  =  ( invg `  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 5746 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3383 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
43sseli 3349 . . . . . . 7  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 12397 . . . . . . 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 16203 . . . . . 6  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 wrdco 12455 . . . . . 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 657 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
111efgrcl 16205 . . . . . 6  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1211simprd 460 . . . . 5  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
1310, 12eleqtrrd 2518 . . . 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 2441 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
171, 14, 15, 16frgpadd 16253 . . . 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 663 . . 3  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  [
( A concat  ( M  o.  (reverse `  A )
) ) ]  .~  )
191, 15efger 16208 . . . . 5  |-  .~  Er  W
2019a1i 11 . . . 4  |-  ( A  e.  W  ->  .~  Er  W )
21 eqid 2441 . . . . 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 16217 . . . 4  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
2320, 22erthi 7143 . . 3  |-  ( A  e.  W  ->  [ ( A concat  ( M  o.  (reverse `  A ) ) ) ]  .~  =  [ (/) ]  .~  )
2414, 15frgp0 16250 . . . . . 6  |-  ( I  e.  _V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2524adantr 462 . . . . 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 460 . . 3  |-  ( A  e.  W  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
2818, 23, 273eqtrd 2477 . 2  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) )
2926simpld 456 . . 3  |-  ( A  e.  W  ->  G  e.  Grp )
30 eqid 2441 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
3114, 15, 1, 30frgpeccl 16251 . . 3  |-  ( A  e.  W  ->  [ A ]  .~  e.  ( Base `  G ) )
3214, 15, 1, 30frgpeccl 16251 . . . 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 2441 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
35 frgpinv.n . . . 4  |-  N  =  ( invg `  G )
3630, 16, 34, 35grpinvid1 15579 . . 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 1213 . 2  |-  ( A  e.  W  ->  (
( N `  [ A ]  .~  )  =  [ ( M  o.  (reverse `  A ) ) ]  .~  <->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) ) )
3828, 37mpbird 232 1  |-  ( A  e.  W  ->  ( N `  [ A ]  .~  )  =  [
( M  o.  (reverse `  A ) ) ]  .~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322   (/)c0 3634   <.cop 3880   <.cotp 3882    e. cmpt 4347    _I cid 4627    X. cxp 4834    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1oc1o 6909   2oc2o 6910    Er wer 7094   [cec 7095   0cc0 9278   ...cfz 11433   #chash 12099  Word cword 12217   concat cconcat 12219   splice csplice 12222  reversecreverse 12223   <"cs2 12464   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407   ~FG cefg 16196  freeGrpcfrgp 16197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-splice 12230  df-reverse 12231  df-s2 12471  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-0g 14376  df-imas 14442  df-divs 14443  df-mnd 15411  df-frmd 15520  df-grp 15538  df-minusg 15539  df-efg 16199  df-frgp 16200
This theorem is referenced by:  vrgpinv  16259
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