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Theorem frgpinv 15359
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 5751 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3346 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
43sseli 3312 . . . . . . 7  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 11756 . . . . . . 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 15308 . . . . . 6  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 wrdco 11763 . . . . . 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 644 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
111efgrcl 15310 . . . . . 6  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1211simprd 450 . . . . 5  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
1310, 12eleqtrrd 2489 . . . 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 2412 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
171, 14, 15, 16frgpadd 15358 . . . 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 650 . . 3  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  [
( A concat  ( M  o.  (reverse `  A )
) ) ]  .~  )
191, 15efger 15313 . . . . 5  |-  .~  Er  W
2019a1i 11 . . . 4  |-  ( A  e.  W  ->  .~  Er  W )
21 eqid 2412 . . . . 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 15322 . . . 4  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
2320, 22erthi 6918 . . 3  |-  ( A  e.  W  ->  [ ( A concat  ( M  o.  (reverse `  A ) ) ) ]  .~  =  [ (/) ]  .~  )
2414, 15frgp0 15355 . . . . . 6  |-  ( I  e.  _V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2524adantr 452 . . . . 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 450 . . 3  |-  ( A  e.  W  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
2818, 23, 273eqtrd 2448 . 2  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) )
2926simpld 446 . . 3  |-  ( A  e.  W  ->  G  e.  Grp )
30 eqid 2412 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
3114, 15, 1, 30frgpeccl 15356 . . 3  |-  ( A  e.  W  ->  [ A ]  .~  e.  ( Base `  G ) )
3214, 15, 1, 30frgpeccl 15356 . . . 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 2412 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
35 frgpinv.n . . . 4  |-  N  =  ( inv g `  G )
3630, 16, 34, 35grpinvid1 14816 . . 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 1184 . 2  |-  ( A  e.  W  ->  (
( N `  [ A ]  .~  )  =  [ ( M  o.  (reverse `  A ) ) ]  .~  <->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) ) )
3828, 37mpbird 224 1  |-  ( A  e.  W  ->  ( N `  [ A ]  .~  )  =  [
( M  o.  (reverse `  A ) ) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    \ cdif 3285   (/)c0 3596   <.cop 3785   <.cotp 3786    e. cmpt 4234    _I cid 4461    X. cxp 4843    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   1oc1o 6684   2oc2o 6685    Er wer 6869   [cec 6870   0cc0 8954   ...cfz 11007   #chash 11581  Word cword 11680   concat cconcat 11681   splice csplice 11684  reversecreverse 11685   <"cs2 11768   Basecbs 13432   +g cplusg 13492   0gc0g 13686   Grpcgrp 14648   inv gcminusg 14649   ~FG cefg 15301  freeGrpcfrgp 15302
This theorem is referenced by:  vrgpinv  15364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-ot 3792  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-fzo 11099  df-hash 11582  df-word 11686  df-concat 11687  df-s1 11688  df-substr 11689  df-splice 11690  df-reverse 11691  df-s2 11775  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-0g 13690  df-imas 13697  df-divs 13698  df-mnd 14653  df-frmd 14757  df-grp 14775  df-minusg 14776  df-efg 15304  df-frgp 15305
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