Theorem plusffn 18579

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 2726	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		plusffn.2	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
4	1, 2, 3	plusffval 18576	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦))
5		ovex 7437	. 2 ⊢ (𝑥(+g‘𝐺)𝑦) ∈ V
6	4, 5	fnmpoi 8052	1 ⊢ ⨣ Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1533   × cxp 5667   Fn wfn 6531  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  +_gcplusg 17203  +_𝑓cplusf 18567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-plusf 18569

This theorem is referenced by:  lmodfopnelem1  20741  tmdcn2  23943  plusfreseq  47096