Theorem plusffn 18316

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 2739	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		plusffn.2	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
4	1, 2, 3	plusffval 18313	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦))
5		ovex 7301	. 2 ⊢ (𝑥(+g‘𝐺)𝑦) ∈ V
6	4, 5	fnmpoi 7896	1 ⊢ ⨣ Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1541   × cxp 5586   Fn wfn 6425  ‘cfv 6430  (class class class)co 7268  Basecbs 16893  +_gcplusg 16943  +_𝑓cplusf 18304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-plusf 18306

This theorem is referenced by:  lmodfopnelem1  20140  tmdcn2  23221  plusfreseq  45278