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Theorem efgtval 16200
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 16199 . . . . 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 456 . . . 4  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76oveqd 6097 . . 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 4056 . . . . . 6  |-  ( a  =  N  ->  <. a ,  a ,  <" b ( M `  b ) "> >.  =  <. N ,  a ,  <" b ( M `  b ) "> >. )
9 oteq2 4057 . . . . . 6  |-  ( a  =  N  ->  <. N , 
a ,  <" b
( M `  b
) "> >.  =  <. N ,  N ,  <" b ( M `  b ) "> >.
)
108, 9eqtrd 2465 . . . . 5  |-  ( a  =  N  ->  <. a ,  a ,  <" b ( M `  b ) "> >.  =  <. N ,  N ,  <" b ( M `  b ) "> >. )
1110oveq2d 6096 . . . 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 5679 . . . . . . 7  |-  ( b  =  A  ->  ( M `  b )  =  ( M `  A ) )
1412, 13s2eqd 12473 . . . . . 6  |-  ( b  =  A  ->  <" b
( M `  b
) ">  =  <" A ( M `
 A ) "> )
1514oteq3d 4061 . . . . 5  |-  ( b  =  A  ->  <. N ,  N ,  <" b
( M `  b
) "> >.  =  <. N ,  N ,  <" A ( M `  A ) "> >.
)
1615oveq2d 6096 . . . 4  |-  ( b  =  A  ->  ( X splice  <. N ,  N ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. N ,  N ,  <" A
( M `  A
) "> >. )
)
17 eqid 2433 . . . 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 6105 . . . 4  |-  ( X splice  <. N ,  N ,  <" A ( M `
 A ) "> >. )  e.  _V
1911, 16, 17, 18ovmpt2 6215 . . 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 2485 . 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 1176 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 958    = wceq 1362    e. wcel 1755    \ cdif 3313   <.cop 3871   <.cotp 3873    e. cmpt 4338    _I cid 4618    X. cxp 4825   -->wf 5402   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1oc1o 6901   2oc2o 6902   0cc0 9270   ...cfz 11424   #chash 12087  Word cword 12205   splice csplice 12210   <"cs2 12452   ~FG cefg 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-ot 3874  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-hash 12088  df-word 12213  df-concat 12215  df-s1 12216  df-substr 12217  df-splice 12218  df-s2 12459
This theorem is referenced by:  efginvrel2  16204  efgredleme  16220  efgredlemc  16222  efgcpbllemb  16232
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