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Theorem efgtval 16324
Description: Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-Sep-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
efgtval  |-  ( ( X  e.  W  /\  N  e.  ( 0 ... ( # `  X
) )  /\  A  e.  ( I  X.  2o ) )  ->  ( N ( T `  X ) A )  =  ( X splice  <. N ,  N ,  <" A
( M `  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)    N( y, z, w, v, n)    X( y, z, w, v, n)

Proof of Theorem efgtval
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . 6  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . . 6  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . . 6  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
51, 2, 3, 4efgtf 16323 . . . . 5  |-  ( X  e.  W  ->  (
( T `  X
)  =  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  /\  ( T `  X ) : ( ( 0 ... ( # `  X
) )  X.  (
I  X.  2o ) ) --> W ) )
65simpld 459 . . . 4  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76oveqd 6207 . . 3  |-  ( X  e.  W  ->  ( N ( T `  X ) A )  =  ( N ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) ) A ) )
8 oteq1 4166 . . . . . 6  |-  ( a  =  N  ->  <. a ,  a ,  <" b ( M `  b ) "> >.  =  <. N ,  a ,  <" b ( M `  b ) "> >. )
9 oteq2 4167 . . . . . 6  |-  ( a  =  N  ->  <. N , 
a ,  <" b
( M `  b
) "> >.  =  <. N ,  N ,  <" b ( M `  b ) "> >.
)
108, 9eqtrd 2492 . . . . 5  |-  ( a  =  N  ->  <. a ,  a ,  <" b ( M `  b ) "> >.  =  <. N ,  N ,  <" b ( M `  b ) "> >. )
1110oveq2d 6206 . . . 4  |-  ( a  =  N  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. N ,  N ,  <" b
( M `  b
) "> >. )
)
12 id 22 . . . . . . 7  |-  ( b  =  A  ->  b  =  A )
13 fveq2 5789 . . . . . . 7  |-  ( b  =  A  ->  ( M `  b )  =  ( M `  A ) )
1412, 13s2eqd 12591 . . . . . 6  |-  ( b  =  A  ->  <" b
( M `  b
) ">  =  <" A ( M `
 A ) "> )
1514oteq3d 4171 . . . . 5  |-  ( b  =  A  ->  <. N ,  N ,  <" b
( M `  b
) "> >.  =  <. N ,  N ,  <" A ( M `  A ) "> >.
)
1615oveq2d 6206 . . . 4  |-  ( b  =  A  ->  ( X splice  <. N ,  N ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. N ,  N ,  <" A
( M `  A
) "> >. )
)
17 eqid 2451 . . . 4  |-  ( a  e.  ( 0 ... ( # `  X
) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a , 
<" b ( M `
 b ) "> >. ) )  =  ( a  e.  ( 0 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )
)
18 ovex 6215 . . . 4  |-  ( X splice  <. N ,  N ,  <" A ( M `
 A ) "> >. )  e.  _V
1911, 16, 17, 18ovmpt2 6326 . . 3  |-  ( ( N  e.  ( 0 ... ( # `  X
) )  /\  A  e.  ( I  X.  2o ) )  ->  ( N ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) A )  =  ( X splice  <. N ,  N ,  <" A
( M `  A
) "> >. )
)
207, 19sylan9eq 2512 . 2  |-  ( ( X  e.  W  /\  ( N  e.  (
0 ... ( # `  X
) )  /\  A  e.  ( I  X.  2o ) ) )  -> 
( N ( T `
 X ) A )  =  ( X splice  <. N ,  N ,  <" A ( M `
 A ) "> >. ) )
21203impb 1184 1  |-  ( ( X  e.  W  /\  N  e.  ( 0 ... ( # `  X
) )  /\  A  e.  ( I  X.  2o ) )  ->  ( N ( T `  X ) A )  =  ( X splice  <. N ,  N ,  <" A
( M `  A
) "> >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    \ cdif 3423   <.cop 3981   <.cotp 3983    |-> cmpt 4448    _I cid 4729    X. cxp 4936   -->wf 5512   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   1oc1o 7013   2oc2o 7014   0cc0 9383   ...cfz 11538   #chash 12204  Word cword 12323   splice csplice 12328   <"cs2 12570   ~FG cefg 16307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-hash 12205  df-word 12331  df-concat 12333  df-s1 12334  df-substr 12335  df-splice 12336  df-s2 12577
This theorem is referenced by:  efginvrel2  16328  efgredleme  16344  efgredlemc  16346  efgcpbllemb  16356
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