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Theorem frgpinv 15073
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 5580 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3208 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
43sseli 3176 . . . . . . 7  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 11479 . . . . . . 7  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  A
)  e. Word  ( I  X.  2o ) )
64, 5syl 15 . . . . . 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 15022 . . . . . 6  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 wrdco 11486 . . . . . 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 643 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
111efgrcl 15024 . . . . . 6  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1211simprd 449 . . . . 5  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
1310, 12eleqtrrd 2360 . . . 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 2283 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
171, 14, 15, 16frgpadd 15072 . . . 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 649 . . 3  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  [
( A concat  ( M  o.  (reverse `  A )
) ) ]  .~  )
191, 15efger 15027 . . . . 5  |-  .~  Er  W
2019a1i 10 . . . 4  |-  ( A  e.  W  ->  .~  Er  W )
21 eqid 2283 . . . . 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 15036 . . . 4  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
2320, 22erthi 6706 . . 3  |-  ( A  e.  W  ->  [ ( A concat  ( M  o.  (reverse `  A ) ) ) ]  .~  =  [ (/) ]  .~  )
2414, 15frgp0 15069 . . . . . 6  |-  ( I  e.  _V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2524adantr 451 . . . . 5  |-  ( ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) )  -> 
( G  e.  Grp  /\ 
[ (/) ]  .~  =  ( 0g `  G ) ) )
2611, 25syl 15 . . . 4  |-  ( A  e.  W  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2726simprd 449 . . 3  |-  ( A  e.  W  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
2818, 23, 273eqtrd 2319 . 2  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) )
2926simpld 445 . . 3  |-  ( A  e.  W  ->  G  e.  Grp )
30 eqid 2283 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
3114, 15, 1, 30frgpeccl 15070 . . 3  |-  ( A  e.  W  ->  [ A ]  .~  e.  ( Base `  G ) )
3214, 15, 1, 30frgpeccl 15070 . . . 4  |-  ( ( M  o.  (reverse `  A
) )  e.  W  ->  [ ( M  o.  (reverse `  A ) ) ]  .~  e.  (
Base `  G )
)
3313, 32syl 15 . . 3  |-  ( A  e.  W  ->  [ ( M  o.  (reverse `  A
) ) ]  .~  e.  ( Base `  G
) )
34 eqid 2283 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
35 frgpinv.n . . . 4  |-  N  =  ( inv g `  G )
3630, 16, 34, 35grpinvid1 14530 . . 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 1182 . 2  |-  ( A  e.  W  ->  (
( N `  [ A ]  .~  )  =  [ ( M  o.  (reverse `  A ) ) ]  .~  <->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) ) )
3828, 37mpbird 223 1  |-  ( A  e.  W  ->  ( N `  [ A ]  .~  )  =  [
( M  o.  (reverse `  A ) ) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455   <.cop 3643   <.cotp 3644    e. cmpt 4077    _I cid 4304    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   [cec 6658   0cc0 8737   ...cfz 10782   #chash 11337  Word cword 11403   concat cconcat 11404   splice csplice 11407  reversecreverse 11408   <"cs2 11491   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   ~FG cefg 15015  freeGrpcfrgp 15016
This theorem is referenced by:  vrgpinv  15078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-frmd 14471  df-grp 14489  df-minusg 14490  df-efg 15018  df-frgp 15019
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