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Theorem mulgfn 14813
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 2380 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2380 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2380 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
5 mulgfn.t . . 3  |-  .x.  =  (.g
`  G )
61, 2, 3, 4, 5mulgfval 14811 . 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 5675 . . 3  |-  ( 0g
`  G )  e. 
_V
8 fvex 5675 . . . 4  |-  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n )  e.  _V
9 fvex 5675 . . . 4  |-  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) )  e. 
_V
108, 9ifex 3733 . . 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 3733 . 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 6352 1  |-  .x.  Fn  ( ZZ  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649   ifcif 3675   {csn 3750   class class class wbr 4146    X. cxp 4809    Fn wfn 5382   ` cfv 5387   0cc0 8916   1c1 8917    < clt 9046   -ucneg 9217   NNcn 9925   ZZcz 10207    seq cseq 11243   Basecbs 13389   +g cplusg 13449   0gc0g 13643   inv gcminusg 14606  .gcmg 14609
This theorem is referenced by:  mulgfvi  14814  tgpmulg2  18038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-neg 9219  df-z 10208  df-seq 11244  df-mulg 14735
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