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Theorem mulgfn 14570
Description: Functionality of the group multiple function. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
mulgfn.b  |-  B  =  ( Base `  G
)
mulgfn.t  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
mulgfn  |-  .x.  Fn  ( ZZ  X.  B
)

Proof of Theorem mulgfn
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgfn.b . . 3  |-  B  =  ( Base `  G
)
2 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2283 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2283 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
5 mulgfn.t . . 3  |-  .x.  =  (.g
`  G )
61, 2, 3, 4, 5mulgfval 14568 . 2  |-  .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
n ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) ) ) ) )
7 fvex 5539 . . 3  |-  ( 0g
`  G )  e. 
_V
8 fvex 5539 . . . 4  |-  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n )  e.  _V
9 fvex 5539 . . . 4  |-  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) )  e. 
_V
108, 9ifex 3623 . . 3  |-  if ( 0  <  n ,  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { x }
) ) `  n
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) ) )  e.  _V
117, 10ifex 3623 . 2  |-  if ( n  =  0 ,  ( 0g `  G
) ,  if ( 0  <  n ,  (  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { x }
) ) `  n
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) ) ) )  e.  _V
126, 11fnmpt2i 6193 1  |-  .x.  Fn  ( ZZ  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687    Fn wfn 5250   ` cfv 5255   0cc0 8737   1c1 8738    < clt 8867   -ucneg 9038   NNcn 9746   ZZcz 10024    seq cseq 11046   Basecbs 13148   +g cplusg 13208   0gc0g 13400   inv gcminusg 14363  .gcmg 14366
This theorem is referenced by:  mulgfvi  14571  tgpmulg2  17777
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-inf2 7342  ax-cnex 8793  ax-resscn 8794
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-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-uni 3828  df-iun 3907  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-recs 6388  df-rdg 6423  df-neg 9040  df-z 10025  df-seq 11047  df-mulg 14492
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