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Theorem plusfval 15736
Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
plusffval.1  |-  B  =  ( Base `  G
)
plusffval.2  |-  .+  =  ( +g  `  G )
plusffval.3  |-  .+^  =  ( +f `  G
)
Assertion
Ref Expression
plusfval  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .+^  Y )  =  ( X  .+  Y ) )

Proof of Theorem plusfval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6286 . 2  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  .+  y
)  =  ( X 
.+  Y ) )
2 plusffval.1 . . 3  |-  B  =  ( Base `  G
)
3 plusffval.2 . . 3  |-  .+  =  ( +g  `  G )
4 plusffval.3 . . 3  |-  .+^  =  ( +f `  G
)
52, 3, 4plusffval 15735 . 2  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
6 ovex 6302 . 2  |-  ( X 
.+  Y )  e. 
_V
71, 5, 6ovmpt2a 6410 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .+^  Y )  =  ( X  .+  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   +fcplusf 15720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-plusf 15724
This theorem is referenced by:  cnmpt1plusg  20316  cnmpt2plusg  20317  tmdcn2  20318  tsmsadd  20379  mhmhmeotmd  27533
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