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Theorem mulgfn 17544
Description: Functionality of the group multiple operation. (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 2622 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 eqid 2622 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
4 eqid 2622 . . 3  |-  ( invg `  G )  =  ( invg `  G )
5 mulgfn.t . . 3  |-  .x.  =  (.g
`  G )
61, 2, 3, 4, 5mulgfval 17542 . 2  |-  .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
n ,  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n ) ,  ( ( invg `  G ) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) ) ) ) )
7 fvex 6201 . . 3  |-  ( 0g
`  G )  e. 
_V
8 fvex 6201 . . . 4  |-  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  n )  e.  _V
9 fvex 6201 . . . 4  |-  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) )  e. 
_V
108, 9ifex 4156 . . 3  |-  if ( 0  <  n ,  (  seq 1 ( ( +g  `  G
) ,  ( NN 
X.  { x }
) ) `  n
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) ) )  e.  _V
117, 10ifex 4156 . 2  |-  if ( n  =  0 ,  ( 0g `  G
) ,  if ( 0  <  n ,  (  seq 1 ( ( +g  `  G
) ,  ( NN 
X.  { x }
) ) `  n
) ,  ( ( invg `  G
) `  (  seq 1 ( ( +g  `  G ) ,  ( NN  X.  { x } ) ) `  -u n ) ) ) )  e.  _V
126, 11fnmpt2i 7239 1  |-  .x.  Fn  ( ZZ  X.  B
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1483   ifcif 4086   {csn 4177   class class class wbr 4653    X. cxp 5112    Fn wfn 5883   ` cfv 5888   0cc0 9936   1c1 9937    < clt 10074   -ucneg 10267   NNcn 11020   ZZcz 11377    seqcseq 12801   Basecbs 15857   +g cplusg 15941   0gc0g 16100   invgcminusg 17423  .gcmg 17540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-neg 10269  df-z 11378  df-seq 12802  df-mulg 17541
This theorem is referenced by:  mulgfvi  17545  tgpmulg2  21898
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