Theorem plusffval 18549

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 6878	. . . . . 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 6878	. . . . . . 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 7410	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
9	4, 4, 8	mpoeq123dv 7468	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
10		df-plusf 18542	. . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
11	3	fvexi 6892	. . . . 5 ⊢ 𝐵 ∈ V
12	6	fvexi 6892	. . . . . . 7 ⊢ + ∈ V
13	12	rnex 7885	. . . . . 6 ⊢ ran + ∈ V
14		p0ex 5375	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7716	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
16		df-ov 7396	. . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6908	. . . . . . 7 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
18	16, 17	eqeltri 2828	. . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
19	18	rgen2w 3065	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
20	11, 11, 15, 19	mpoexw 8047	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V
21	9, 10, 20	fvmpt 6984	. . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
22		fvprc 6870	. . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅)
23		fvprc 6870	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	eqtrid 2783	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25	24	olcd 872	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26		0mpo0 7476	. . . . 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 845   = wceq 1541   ∈ wcel 2106  Vcvv 3473   ∪ cun 3942  ∅c0 4318  {csn 4622  ⟨cop 4628  ran crn 5670  ‘cfv 6532  (class class class)co 7393   ∈ cmpo 7395  Basecbs 17126  +_gcplusg 17179  +_𝑓cplusf 18540

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-plusf 18542

This theorem is referenced by:  plusfval  18550  plusfeq  18551  plusffn  18552  mgmplusf  18553  rlmscaf  20779  istgp2  23524  oppgtmd  23530  submtmd  23537  prdstmdd  23557  ressplusf  31998  pl1cn  32766