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Theorem mulgfn 14586
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 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2296 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2296 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
5 mulgfn.t . . 3  |-  .x.  =  (.g
`  G )
61, 2, 3, 4, 5mulgfval 14584 . 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 5555 . . 3  |-  ( 0g
`  G )  e. 
_V
8 fvex 5555 . . . 4  |-  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n )  e.  _V
9 fvex 5555 . . . 4  |-  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) )  e. 
_V
108, 9ifex 3636 . . 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 3636 . 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 6209 1  |-  .x.  Fn  ( ZZ  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703    Fn wfn 5266   ` cfv 5271   0cc0 8753   1c1 8754    < clt 8883   -ucneg 9054   NNcn 9762   ZZcz 10040    seq cseq 11062   Basecbs 13164   +g cplusg 13224   0gc0g 13416   inv gcminusg 14379  .gcmg 14382
This theorem is referenced by:  mulgfvi  14587  tgpmulg2  17793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-neg 9056  df-z 10041  df-seq 11063  df-mulg 14508
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