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Theorem mulgfn 14886
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 2436 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2436 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2436 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
5 mulgfn.t . . 3  |-  .x.  =  (.g
`  G )
61, 2, 3, 4, 5mulgfval 14884 . 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 5735 . . 3  |-  ( 0g
`  G )  e. 
_V
8 fvex 5735 . . . 4  |-  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n )  e.  _V
9 fvex 5735 . . . 4  |-  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) )  e. 
_V
108, 9ifex 3790 . . 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 3790 . 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 6413 1  |-  .x.  Fn  ( ZZ  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   ifcif 3732   {csn 3807   class class class wbr 4205    X. cxp 4869    Fn wfn 5442   ` cfv 5447   0cc0 8983   1c1 8984    < clt 9113   -ucneg 9285   NNcn 9993   ZZcz 10275    seq cseq 11316   Basecbs 13462   +g cplusg 13522   0gc0g 13716   inv gcminusg 14679  .gcmg 14682
This theorem is referenced by:  mulgfvi  14887  tgpmulg2  18117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-recs 6626  df-rdg 6661  df-neg 9287  df-z 10276  df-seq 11317  df-mulg 14808
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