Theorem plusffval 18614

Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)

Hypotheses

Ref	Expression
plusffval.1	⊢ 𝐵 = (Base‘𝐺)
plusffval.2	⊢ + = (+g‘𝐺)
plusffval.3	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
plusffval	⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦

Allowed substitution hints:   ⨣ (𝑥,𝑦)

Proof of Theorem plusffval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plusffval.3	. 2 ⊢ ⨣ = (+𝑓‘𝐺)
2		fveq2 6840	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		plusffval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6840	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		plusffval.2	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2789	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7384	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
9	4, 4, 8	mpoeq123dv 7442	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
10		df-plusf 18607	. . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
11	3	fvexi 6854	. . . . 5 ⊢ 𝐵 ∈ V
12	6	fvexi 6854	. . . . . . 7 ⊢ + ∈ V
13	12	rnex 7861	. . . . . 6 ⊢ ran + ∈ V
14		p0ex 5326	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7698	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
16		df-ov 7370	. . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6868	. . . . . . 7 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
18	16, 17	eqeltri 2832	. . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
19	18	rgen2w 3056	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
20	11, 11, 15, 19	mpoexw 8031	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V
21	9, 10, 20	fvmpt 6947	. . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
22		fvprc 6832	. . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅)
23		fvprc 6832	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	eqtrid 2783	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25	24	olcd 875	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26		0mpo0 7450	. . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = ∅)
27	25, 26	syl 17	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = ∅)
28	22, 27	eqtr4d 2774	. . 3 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
29	21, 28	pm2.61i 182	. 2 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
30	1, 29	eqtri 2759	1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887  ∅c0 4273  {csn 4567  ⟨cop 4573  ran crn 5632  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  Basecbs 17179  +_gcplusg 17220  +_𝑓cplusf 18605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-plusf 18607

This theorem is referenced by:  plusfval  18615  plusfeq  18616  plusffn  18617  mgmplusf  18618  rlmscaf  21202  istgp2  24056  oppgtmd  24062  submtmd  24069  prdstmdd  24089  ressplusf  33023  pl1cn  34099