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Theorem efginvrel1 16218
Description: The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
Assertion
Ref Expression
efginvrel1  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  A
)  .~  (/) )
Distinct variable groups:    y, z    v, n, w, y, z   
n, M, v, w   
n, W, v, w, y, z    y,  .~ , z    n, I, v, w, y, z
Allowed substitution hints:    A( y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    M( y, z)

Proof of Theorem efginvrel1
Dummy variables  a 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . . . 10  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 fviss 5746 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3383 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
43sseli 3349 . . . . . . . 8  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 12397 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  A
)  e. Word  ( I  X.  2o ) )
64, 5syl 16 . . . . . . 7  |-  ( A  e.  W  ->  (reverse `  A )  e. Word  (
I  X.  2o ) )
7 efgval2.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
87efgmf 16203 . . . . . . 7  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 revco 12458 . . . . . . 7  |-  ( ( (reverse `  A )  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  (reverse `  ( M  o.  (reverse `  A
) ) ) )
106, 8, 9sylancl 657 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  (reverse `  ( M  o.  (reverse `  A
) ) ) )
11 revrev 12403 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  (reverse `  A ) )  =  A )
124, 11syl 16 . . . . . . 7  |-  ( A  e.  W  ->  (reverse `  (reverse `  A )
)  =  A )
1312coeq2d 4998 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  ( M  o.  A ) )
1410, 13eqtr3d 2475 . . . . 5  |-  ( A  e.  W  ->  (reverse `  ( M  o.  (reverse `  A ) ) )  =  ( M  o.  A ) )
1514coeq2d 4998 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  ( M  o.  ( M  o.  A
) ) )
16 wrdf 12236 . . . . . . . . 9  |-  ( A  e. Word  ( I  X.  2o )  ->  A :
( 0..^ ( # `  A ) ) --> ( I  X.  2o ) )
174, 16syl 16 . . . . . . . 8  |-  ( A  e.  W  ->  A : ( 0..^ (
# `  A )
) --> ( I  X.  2o ) )
1817ffvelrnda 5840 . . . . . . 7  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( A `  c
)  e.  ( I  X.  2o ) )
197efgmnvl 16204 . . . . . . 7  |-  ( ( A `  c )  e.  ( I  X.  2o )  ->  ( M `
 ( M `  ( A `  c ) ) )  =  ( A `  c ) )
2018, 19syl 16 . . . . . 6  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( M `  ( M `  ( A `  c ) ) )  =  ( A `  c ) )
2120mpteq2dva 4375 . . . . 5  |-  ( A  e.  W  ->  (
c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( A `
 c ) ) )
228ffvelrni 5839 . . . . . . 7  |-  ( ( A `  c )  e.  ( I  X.  2o )  ->  ( M `
 ( A `  c ) )  e.  ( I  X.  2o ) )
2318, 22syl 16 . . . . . 6  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( M `  ( A `  c )
)  e.  ( I  X.  2o ) )
24 fcompt 5876 . . . . . . 7  |-  ( ( M : ( I  X.  2o ) --> ( I  X.  2o )  /\  A : ( 0..^ ( # `  A
) ) --> ( I  X.  2o ) )  ->  ( M  o.  A )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( A `  c ) ) ) )
258, 17, 24sylancr 658 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  A )  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( M `  ( A `
 c ) ) ) )
268a1i 11 . . . . . . 7  |-  ( A  e.  W  ->  M : ( I  X.  2o ) --> ( I  X.  2o ) )
2726feqmptd 5741 . . . . . 6  |-  ( A  e.  W  ->  M  =  ( a  e.  ( I  X.  2o )  |->  ( M `  a ) ) )
28 fveq2 5688 . . . . . 6  |-  ( a  =  ( M `  ( A `  c ) )  ->  ( M `  a )  =  ( M `  ( M `
 ( A `  c ) ) ) )
2923, 25, 27, 28fmptco 5873 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) ) )
3017feqmptd 5741 . . . . 5  |-  ( A  e.  W  ->  A  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( A `  c ) ) )
3121, 29, 303eqtr4d 2483 . . . 4  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  A )
3215, 31eqtrd 2473 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  A )
3332oveq2d 6106 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  =  ( ( M  o.  (reverse `  A
) ) concat  A )
)
34 wrdco 12455 . . . . 5  |-  ( ( (reverse `  A )  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
356, 8, 34sylancl 657 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
361efgrcl 16205 . . . . 5  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
3736simprd 460 . . . 4  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
3835, 37eleqtrrd 2518 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e.  W
)
39 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
40 efgval2.t . . . 4  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
411, 39, 7, 40efginvrel2 16217 . . 3  |-  ( ( M  o.  (reverse `  A
) )  e.  W  ->  ( ( M  o.  (reverse `  A ) ) concat 
( M  o.  (reverse `  ( M  o.  (reverse `  A ) ) ) ) )  .~  (/) )
4238, 41syl 16 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  .~  (/) )
4333, 42eqbrtrrd 4311 1  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  A
)  .~  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322   (/)c0 3634   <.cop 3880   <.cotp 3882   class class class wbr 4289    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   0cc0 9278   ...cfz 11433  ..^cfzo 11544   #chash 12099  Word cword 12217   concat cconcat 12219   splice csplice 12222  reversecreverse 12223   <"cs2 12464   ~FG cefg 16196
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-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-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  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-n0 10576  df-z 10643  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-efg 16199
This theorem is referenced by:  frgp0  16250
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