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Theorem efginvrel1 16225
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 5749 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3386 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
43sseli 3352 . . . . . . . 8  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 12401 . . . . . . . 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 16210 . . . . . . 7  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 revco 12462 . . . . . . 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 662 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  (reverse `  ( M  o.  (reverse `  A
) ) ) )
11 revrev 12407 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  (reverse `  A ) )  =  A )
124, 11syl 16 . . . . . . 7  |-  ( A  e.  W  ->  (reverse `  (reverse `  A )
)  =  A )
1312coeq2d 5002 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  ( M  o.  A ) )
1410, 13eqtr3d 2477 . . . . 5  |-  ( A  e.  W  ->  (reverse `  ( M  o.  (reverse `  A ) ) )  =  ( M  o.  A ) )
1514coeq2d 5002 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  ( M  o.  ( M  o.  A
) ) )
16 wrdf 12240 . . . . . . . . 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 5843 . . . . . . 7  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( A `  c
)  e.  ( I  X.  2o ) )
197efgmnvl 16211 . . . . . . 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 4378 . . . . 5  |-  ( A  e.  W  ->  (
c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( A `
 c ) ) )
228ffvelrni 5842 . . . . . . 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 5879 . . . . . . 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 663 . . . . . 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 5744 . . . . . 6  |-  ( A  e.  W  ->  M  =  ( a  e.  ( I  X.  2o )  |->  ( M `  a ) ) )
28 fveq2 5691 . . . . . 6  |-  ( a  =  ( M `  ( A `  c ) )  ->  ( M `  a )  =  ( M `  ( M `
 ( A `  c ) ) ) )
2923, 25, 27, 28fmptco 5876 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) ) )
3017feqmptd 5744 . . . . 5  |-  ( A  e.  W  ->  A  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( A `  c ) ) )
3121, 29, 303eqtr4d 2485 . . . 4  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  A )
3215, 31eqtrd 2475 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  A )
3332oveq2d 6107 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  =  ( ( M  o.  (reverse `  A
) ) concat  A )
)
34 wrdco 12459 . . . . 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 662 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
361efgrcl 16212 . . . . 5  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
3736simprd 463 . . . 4  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
3835, 37eleqtrrd 2520 . . 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 16224 . . 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 4314 1  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  A
)  .~  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972    \ cdif 3325   (/)c0 3637   <.cop 3883   <.cotp 3885   class class class wbr 4292    e. cmpt 4350    _I cid 4631    X. cxp 4838    o. ccom 4844   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1oc1o 6913   2oc2o 6914   0cc0 9282   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   concat cconcat 12223   splice csplice 12226  reversecreverse 12227   <"cs2 12468   ~FG cefg 16203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-reverse 12235  df-s2 12475  df-efg 16206
This theorem is referenced by:  frgp0  16257
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