Theorem plusffn 17073

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	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
plusffn	⊢ ⨣ Fn (𝐵 × 𝐵)

Proof of Theorem plusffn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plusffn.1	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2610	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		plusffn.2	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
4	1, 2, 3	plusffval 17070	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦))
5		ovex 6577	. 2 ⊢ (𝑥(+g‘𝐺)𝑦) ∈ V
6	4, 5	fnmpt2i 7128	1 ⊢ ⨣ Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1475   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  +_𝑓cplusf 17062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-plusf 17064

This theorem is referenced by:  lmodfopnelem1  18722  tmdcn2  21703  plusfreseq  41562