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Theorem efginvrel1 15362
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 5786 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
31, 2eqsstri 3380 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
43sseli 3346 . . . . . . . 8  |-  ( A  e.  W  ->  A  e. Word  ( I  X.  2o ) )
5 revcl 11795 . . . . . . . 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 15347 . . . . . . 7  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9 revco 11805 . . . . . . 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 645 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  (reverse `  ( M  o.  (reverse `  A
) ) ) )
11 revrev 11801 . . . . . . . 8  |-  ( A  e. Word  ( I  X.  2o )  ->  (reverse `  (reverse `  A ) )  =  A )
124, 11syl 16 . . . . . . 7  |-  ( A  e.  W  ->  (reverse `  (reverse `  A )
)  =  A )
1312coeq2d 5037 . . . . . 6  |-  ( A  e.  W  ->  ( M  o.  (reverse `  (reverse `  A ) ) )  =  ( M  o.  A ) )
1410, 13eqtr3d 2472 . . . . 5  |-  ( A  e.  W  ->  (reverse `  ( M  o.  (reverse `  A ) ) )  =  ( M  o.  A ) )
1514coeq2d 5037 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  ( M  o.  ( M  o.  A
) ) )
16 wrdf 11735 . . . . . . . . 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 5872 . . . . . . 7  |-  ( ( A  e.  W  /\  c  e.  ( 0..^ ( # `  A
) ) )  -> 
( A `  c
)  e.  ( I  X.  2o ) )
197efgmnvl 15348 . . . . . . 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 4297 . . . . 5  |-  ( A  e.  W  ->  (
c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( A `
 c ) ) )
228ffvelrni 5871 . . . . . . 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 5906 . . . . . . 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 646 . . . . . 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 5781 . . . . . 6  |-  ( A  e.  W  ->  M  =  ( a  e.  ( I  X.  2o )  |->  ( M `  a ) ) )
28 fveq2 5730 . . . . . 6  |-  ( a  =  ( M `  ( A `  c ) )  ->  ( M `  a )  =  ( M `  ( M `
 ( A `  c ) ) ) )
2923, 25, 27, 28fmptco 5903 . . . . 5  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  ( c  e.  ( 0..^ ( # `  A
) )  |->  ( M `
 ( M `  ( A `  c ) ) ) ) )
3017feqmptd 5781 . . . . 5  |-  ( A  e.  W  ->  A  =  ( c  e.  ( 0..^ ( # `  A ) )  |->  ( A `  c ) ) )
3121, 29, 303eqtr4d 2480 . . . 4  |-  ( A  e.  W  ->  ( M  o.  ( M  o.  A ) )  =  A )
3215, 31eqtrd 2470 . . 3  |-  ( A  e.  W  ->  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) )  =  A )
3332oveq2d 6099 . 2  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  ( M  o.  (reverse `  ( M  o.  (reverse `  A
) ) ) ) )  =  ( ( M  o.  (reverse `  A
) ) concat  A )
)
34 wrdco 11802 . . . . 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 645 . . . 4  |-  ( A  e.  W  ->  ( M  o.  (reverse `  A
) )  e. Word  (
I  X.  2o ) )
361efgrcl 15349 . . . . 5  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
3736simprd 451 . . . 4  |-  ( A  e.  W  ->  W  = Word  ( I  X.  2o ) )
3835, 37eleqtrrd 2515 . . 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 15361 . . 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 4236 1  |-  ( A  e.  W  ->  (
( M  o.  (reverse `  A ) ) concat  A
)  .~  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319   (/)c0 3630   <.cop 3819   <.cotp 3820   class class class wbr 4214    e. cmpt 4268    _I cid 4495    X. cxp 4878    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720   0cc0 8992   ...cfz 11045  ..^cfzo 11137   #chash 11620  Word cword 11719   concat cconcat 11720   splice csplice 11723  reversecreverse 11724   <"cs2 11807   ~FG cefg 15340
This theorem is referenced by:  frgp0  15394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-reverse 11730  df-s2 11814  df-efg 15343
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