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Theorem plusffn 14634
Description: The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
mndplusf.1  |-  B  =  ( Base `  G
)
mndplusf.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 mndplusf.1 . . 3  |-  B  =  ( Base `  G
)
2 eqid 2389 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
3 mndplusf.2 . . 3  |-  .+^  =  ( + f `  G
)
41, 2, 3plusffval 14631 . 2  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x ( +g  `  G
) y ) )
5 ovex 6047 . 2  |-  ( x ( +g  `  G
) y )  e. 
_V
64, 5fnmpt2i 6361 1  |-  .+^  Fn  ( B  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    X. cxp 4818    Fn wfn 5391   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   + fcplusf 14616
This theorem is referenced by:  tmdcn2  18042
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-plusf 14620
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