MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpinv Structured version   Unicode version

Theorem frgpinv 16655
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 5932 . . . . . . . . 9  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3539 . . . . . . . 8  |-  W  C_ Word  ( I  X.  2o )
43sseli 3505 . . . . . . 7  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 12715 . . . . . . 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 16604 . . . . . 6  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 wrdco 12777 . . . . . 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 662 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
111efgrcl 16606 . . . . . 6  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
1211simprd 463 . . . . 5  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
1310, 12eleqtrrd 2558 . . . 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 2467 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
171, 14, 15, 16frgpadd 16654 . . . 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 668 . . 3  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  [
( A concat  ( M  o.  (reverse `  A )
) ) ]  .~  )
191, 15efger 16609 . . . . 5  |-  .~  Er  W
2019a1i 11 . . . 4  |-  ( A  e.  W  ->  .~  Er  W )
21 eqid 2467 . . . . 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 16618 . . . 4  |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A ) ) )  .~  (/) )
2320, 22erthi 7370 . . 3  |-  ( A  e.  W  ->  [ ( A concat  ( M  o.  (reverse `  A ) ) ) ]  .~  =  [ (/) ]  .~  )
2414, 15frgp0 16651 . . . . . 6  |-  ( I  e.  _V  ->  ( G  e.  Grp  /\  [ (/)
]  .~  =  ( 0g `  G ) ) )
2524adantr 465 . . . . 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 463 . . 3  |-  ( A  e.  W  ->  [ (/) ]  .~  =  ( 0g
`  G ) )
2818, 23, 273eqtrd 2512 . 2  |-  ( A  e.  W  ->  ( [ A ]  .~  ( +g  `  G ) [ ( M  o.  (reverse `  A ) ) ]  .~  )  =  ( 0g `  G ) )
2926simpld 459 . . 3  |-  ( A  e.  W  ->  G  e.  Grp )
30 eqid 2467 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
3114, 15, 1, 30frgpeccl 16652 . . 3  |-  ( A  e.  W  ->  [ A ]  .~  e.  ( Base `  G ) )
3214, 15, 1, 30frgpeccl 16652 . . . 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 2467 . . . 4  |-  ( 0g
`  G )  =  ( 0g `  G
)
35 frgpinv.n . . . 4  |-  N  =  ( invg `  G )
3630, 16, 34, 35grpinvid1 15970 . . 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 1228 . 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 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478   (/)c0 3790   <.cop 4039   <.cotp 4041    |-> cmpt 4511    _I cid 4796    X. cxp 5003    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1oc1o 7135   2oc2o 7136    Er wer 7320   [cec 7321   0cc0 9504   ...cfz 11684   #chash 12385  Word cword 12515   concat cconcat 12517   splice csplice 12520  reversecreverse 12521   <"cs2 12786   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926   ~FG cefg 16597  freeGrpcfrgp 16598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12793  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-0g 14714  df-imas 14780  df-qus 14781  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-frmd 15889  df-grp 15929  df-minusg 15930  df-efg 16600  df-frgp 16601
This theorem is referenced by:  vrgpinv  16660
  Copyright terms: Public domain W3C validator