MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgtval Structured version   Unicode version

Theorem efgtval 15356
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 15355 . . . . 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 447 . . . 4  |-  ( X  e.  W  ->  ( T `  X )  =  ( a  e.  ( 0 ... ( # `
 X ) ) ,  b  e.  ( I  X.  2o ) 
|->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >.
) ) )
76oveqd 6099 . . 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 3994 . . . . . 6  |-  ( a  =  N  ->  <. a ,  a ,  <" b ( M `  b ) "> >.  =  <. N ,  a ,  <" b ( M `  b ) "> >. )
9 oteq2 3995 . . . . . 6  |-  ( a  =  N  ->  <. N , 
a ,  <" b
( M `  b
) "> >.  =  <. N ,  N ,  <" b ( M `  b ) "> >.
)
108, 9eqtrd 2469 . . . . 5  |-  ( a  =  N  ->  <. a ,  a ,  <" b ( M `  b ) "> >.  =  <. N ,  N ,  <" b ( M `  b ) "> >. )
1110oveq2d 6098 . . . 4  |-  ( a  =  N  ->  ( X splice  <. a ,  a ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. N ,  N ,  <" b
( M `  b
) "> >. )
)
12 id 21 . . . . . . 7  |-  ( b  =  A  ->  b  =  A )
13 fveq2 5729 . . . . . . 7  |-  ( b  =  A  ->  ( M `  b )  =  ( M `  A ) )
1412, 13s2eqd 11827 . . . . . 6  |-  ( b  =  A  ->  <" b
( M `  b
) ">  =  <" A ( M `
 A ) "> )
1514oteq3d 3999 . . . . 5  |-  ( b  =  A  ->  <. N ,  N ,  <" b
( M `  b
) "> >.  =  <. N ,  N ,  <" A ( M `  A ) "> >.
)
1615oveq2d 6098 . . . 4  |-  ( b  =  A  ->  ( X splice  <. N ,  N ,  <" b ( M `  b ) "> >. )  =  ( X splice  <. N ,  N ,  <" A
( M `  A
) "> >. )
)
17 eqid 2437 . . . 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 6107 . . . 4  |-  ( X splice  <. N ,  N ,  <" A ( M `
 A ) "> >. )  e.  _V
1911, 16, 17, 18ovmpt2 6210 . . 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 2489 . 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 1150 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3318   <.cop 3818   <.cotp 3819    e. cmpt 4267    _I cid 4494    X. cxp 4877   -->wf 5451   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   1oc1o 6718   2oc2o 6719   0cc0 8991   ...cfz 11044   #chash 11619  Word cword 11718   splice csplice 11722   <"cs2 11806   ~FG cefg 15339
This theorem is referenced by:  efginvrel2  15360  efgredleme  15376  efgredlemc  15378  efgcpbllemb  15388
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-ot 3825  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-fzo 11137  df-hash 11620  df-word 11724  df-concat 11725  df-s1 11726  df-substr 11727  df-splice 11728  df-s2 11813
  Copyright terms: Public domain W3C validator