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Theorem plusffn 16204
Description: The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
plusffn.1  |-  B  =  ( Base `  G
)
plusffn.2  |-  .+^  =  ( +f `  G
)
Assertion
Ref Expression
plusffn  |-  .+^  Fn  ( B  X.  B )

Proof of Theorem plusffn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plusffn.1 . . 3  |-  B  =  ( Base `  G
)
2 eqid 2402 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 plusffn.2 . . 3  |-  .+^  =  ( +f `  G
)
41, 2, 3plusffval 16201 . 2  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x ( +g  `  G
) y ) )
5 ovex 6306 . 2  |-  ( x ( +g  `  G
) y )  e. 
_V
64, 5fnmpt2i 6853 1  |-  .+^  Fn  ( B  X.  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    X. cxp 4821    Fn wfn 5564   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   +fcplusf 16193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-plusf 16195
This theorem is referenced by:  tmdcn2  20880  plusfreseq  38089
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