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Theorem efginvrel1 15037
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 5580 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3208 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
43sseli 3176 . . . . . . . 8  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 11479 . . . . . . . 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 15022 . . . . . . 7  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 revco 11489 . . . . . . 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 11485 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  (reverse `  A ) )  =  A )
124, 11syl 15 . . . . . . 7  |-  ( A  e.  W  ->  (reverse `  (reverse `  A )
)  =  A )
1312coeq2d 4846 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  ( M  o.  A ) )
1410, 13eqtr3d 2317 . . . . 5  |-  ( A  e.  W  ->  (reverse `  ( M  o.  (reverse `  A ) ) )  =  ( M  o.  A ) )
1514coeq2d 4846 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  ( M  o.  ( M  o.  A
) ) )
16 wrdf 11419 . . . . . . . . 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 5663 . . . . . . . 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 15023 . . . . . . 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 4106 . . . . 5  |-  ( A  e.  W  ->  (
c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( A `
 c ) ) )
238ffvelrni 5664 . . . . . . 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 5694 . . . . . . 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 5575 . . . . . 6  |-  ( A  e.  W  ->  M  =  ( a  e.  ( I  X.  2o )  |->  ( M `  a ) ) )
29 fveq2 5525 . . . . . 6  |-  ( a  =  ( M `  ( A `  c ) )  ->  ( M `  a )  =  ( M `  ( M `
 ( A `  c ) ) ) )
3024, 26, 28, 29fmptco 5691 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) ) )
3117feqmptd 5575 . . . . 5  |-  ( A  e.  W  ->  A  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( A `  c ) ) )
3222, 30, 313eqtr4d 2325 . . . 4  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  A )
3315, 32eqtrd 2315 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  A )
3433oveq2d 5874 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  =  ( ( M  o.  (reverse `  A
) ) concat  A )
)
35 wrdco 11486 . . . . 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 15024 . . . . 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 2360 . . 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 15036 . . 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 4045 1  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  A
)  .~  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   (/)c0 3455   <.cop 3643   <.cotp 3644   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   concat cconcat 11404   splice csplice 11407  reversecreverse 11408   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  frgp0  15069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-efg 15018
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