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Theorem plusfval 17017
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 + = (+g𝐺)
plusffval.3 = (+𝑓𝐺)
Assertion
Ref Expression
plusfval ((𝑋𝐵𝑌𝐵) → (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 6536 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 + 𝑦) = (𝑋 + 𝑌))
2 plusffval.1 . . 3 𝐵 = (Base‘𝐺)
3 plusffval.2 . . 3 + = (+g𝐺)
4 plusffval.3 . . 3 = (+𝑓𝐺)
52, 3, 4plusffval 17016 . 2 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
6 ovex 6555 . 2 (𝑋 + 𝑌) ∈ V
71, 5, 6ovmpt2a 6667 1 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋 + 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  +𝑓cplusf 17008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-plusf 17010
This theorem is referenced by:  mndpfo  17083  lmodfopne  18670  cnmpt1plusg  21643  cnmpt2plusg  21644  tmdcn2  21645  tsmsadd  21702  mhmhmeotmd  29107  plusfreseq  41557
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