Theorem plusffn 18687

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 2740	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		plusffn.2	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
4	1, 2, 3	plusffval 18684	. 2 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥(+g‘𝐺)𝑦))
5		ovex 7481	. 2 ⊢ (𝑥(+g‘𝐺)𝑦) ∈ V
6	4, 5	fnmpoi 8111	1 ⊢ ⨣ Fn (𝐵 × 𝐵)

Syntax hints:   = wceq 1537   × cxp 5698   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  +_𝑓cplusf 18675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-plusf 18677

This theorem is referenced by:  lmodfopnelem1  20918  tmdcn2  24118  plusfreseq  47887