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Theorem plusffn 17019
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 = (+𝑓𝐺)
Assertion
Ref Expression
plusffn Fn (𝐵 × 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 2609 . . 3 (+g𝐺) = (+g𝐺)
3 plusffn.2 . . 3 = (+𝑓𝐺)
41, 2, 3plusffval 17016 . 2 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥(+g𝐺)𝑦))
5 ovex 6555 . 2 (𝑥(+g𝐺)𝑦) ∈ V
64, 5fnmpt2i 7105 1 Fn (𝐵 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474   × cxp 5026   Fn wfn 5785  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:  lmodfopnelem1  18668  tmdcn2  21645  plusfreseq  41557
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