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Theorem plusfval 15752
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 6290 . 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 15751 . 2  |-  .+^  =  ( x  e.  B , 
y  e.  B  |->  ( x  .+  y ) )
6 ovex 6309 . 2  |-  ( X 
.+  Y )  e. 
_V
71, 5, 6ovmpt2a 6418 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .+^  Y )  =  ( X  .+  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   +fcplusf 15743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-plusf 15745
This theorem is referenced by:  mndpfo  15818  cnmpt1plusg  20459  cnmpt2plusg  20460  tmdcn2  20461  tsmsadd  20522  mhmhmeotmd  27782  plusfreseq  32298
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