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Theorem efginvrel1 15053
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 5596 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3221 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
43sseli 3189 . . . . . . . 8  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 11495 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  A
)  e. Word  ( I  X.  2o ) )
64, 5syl 15 . . . . . . 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 15038 . . . . . . 7  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 revco 11505 . . . . . . 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 643 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  (reverse `  ( M  o.  (reverse `  A
) ) ) )
11 revrev 11501 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  (reverse `  A ) )  =  A )
124, 11syl 15 . . . . . . 7  |-  ( A  e.  W  ->  (reverse `  (reverse `  A )
)  =  A )
1312coeq2d 4862 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  ( M  o.  A ) )
1410, 13eqtr3d 2330 . . . . 5  |-  ( A  e.  W  ->  (reverse `  ( M  o.  (reverse `  A ) ) )  =  ( M  o.  A ) )
1514coeq2d 4862 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  ( M  o.  ( M  o.  A
) ) )
16 wrdf 11435 . . . . . . . . 9  |-  ( A  e. Word  ( I  X.  2o )  ->  A :
( 0..^ ( # `  A ) ) --> ( I  X.  2o ) )
174, 16syl 15 . . . . . . . 8  |-  ( A  e.  W  ->  A : ( 0..^ (
# `  A )
) --> ( I  X.  2o ) )
18 ffvelrn 5679 . . . . . . . 8  |-  ( ( A : ( 0..^ ( # `  A
) ) --> ( I  X.  2o )  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( A `  c
)  e.  ( I  X.  2o ) )
1917, 18sylan 457 . . . . . . 7  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( A `  c
)  e.  ( I  X.  2o ) )
207efgmnvl 15039 . . . . . . 7  |-  ( ( A `  c )  e.  ( I  X.  2o )  ->  ( M `
 ( M `  ( A `  c ) ) )  =  ( A `  c ) )
2119, 20syl 15 . . . . . 6  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( M `  ( M `  ( A `  c ) ) )  =  ( A `  c ) )
2221mpteq2dva 4122 . . . . 5  |-  ( A  e.  W  ->  (
c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( A `
 c ) ) )
238ffvelrni 5680 . . . . . . 7  |-  ( ( A `  c )  e.  ( I  X.  2o )  ->  ( M `
 ( A `  c ) )  e.  ( I  X.  2o ) )
2419, 23syl 15 . . . . . 6  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( M `  ( A `  c )
)  e.  ( I  X.  2o ) )
25 fcompt 5710 . . . . . . 7  |-  ( ( M : ( I  X.  2o ) --> ( I  X.  2o )  /\  A : ( 0..^ ( # `  A
) ) --> ( I  X.  2o ) )  ->  ( M  o.  A )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( A `  c ) ) ) )
268, 17, 25sylancr 644 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  A )  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( M `  ( A `
 c ) ) ) )
278a1i 10 . . . . . . 7  |-  ( A  e.  W  ->  M : ( I  X.  2o ) --> ( I  X.  2o ) )
2827feqmptd 5591 . . . . . 6  |-  ( A  e.  W  ->  M  =  ( a  e.  ( I  X.  2o )  |->  ( M `  a ) ) )
29 fveq2 5541 . . . . . 6  |-  ( a  =  ( M `  ( A `  c ) )  ->  ( M `  a )  =  ( M `  ( M `
 ( A `  c ) ) ) )
3024, 26, 28, 29fmptco 5707 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) ) )
3117feqmptd 5591 . . . . 5  |-  ( A  e.  W  ->  A  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( A `  c ) ) )
3222, 30, 313eqtr4d 2338 . . . 4  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  A )
3315, 32eqtrd 2328 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  A )
3433oveq2d 5890 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  =  ( ( M  o.  (reverse `  A
) ) concat  A )
)
35 wrdco 11502 . . . . 5  |-  ( ( (reverse `  A )  e. Word  ( I  X.  2o )  /\  M : ( I  X.  2o ) --> ( I  X.  2o ) )  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
366, 8, 35sylancl 643 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
371efgrcl 15040 . . . . 5  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
3837simprd 449 . . . 4  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
3936, 38eleqtrrd 2373 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e.  W
)
40 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
41 efgval2.t . . . 4  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
421, 40, 7, 41efginvrel2 15052 . . 3  |-  ( ( M  o.  (reverse `  A
) )  e.  W  ->  ( ( M  o.  (reverse `  A ) ) concat 
( M  o.  (reverse `  ( M  o.  (reverse `  A ) ) ) ) )  .~  (/) )
4339, 42syl 15 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  .~  (/) )
4434, 43eqbrtrrd 4061 1  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  A
)  .~  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   (/)c0 3468   <.cop 3656   <.cotp 3657   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   concat cconcat 11420   splice csplice 11423  reversecreverse 11424   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  frgp0  15085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-efg 15034
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