Theorem plusffval 18092

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 6706	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		plusffval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2792	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6706	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		plusffval.2	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2792	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7219	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
9	4, 4, 8	mpoeq123dv 7275	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
10		df-plusf 18085	. . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
11	3	fvexi 6720	. . . . 5 ⊢ 𝐵 ∈ V
12	6	fvexi 6720	. . . . . . 7 ⊢ + ∈ V
13	12	rnex 7679	. . . . . 6 ⊢ ran + ∈ V
14		p0ex 5266	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7520	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
16		df-ov 7205	. . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6734	. . . . . . 7 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
18	16, 17	eqeltri 2830	. . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
19	18	rgen2w 3067	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
20	11, 11, 15, 19	mpoexw 7838	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V
21	9, 10, 20	fvmpt 6807	. . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
22		fvprc 6698	. . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅)
23		fvprc 6698	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	syl5eq 2786	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25	24	olcd 874	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26		0mpo0 7283	. . . . 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 2777	. . 3 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
29	21, 28	pm2.61i 185	. 2 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
30	1, 29	eqtri 2762	1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1543   ∈ wcel 2110  Vcvv 3401   ∪ cun 3855  ∅c0 4227  {csn 4531  ⟨cop 4537  ran crn 5541  ‘cfv 6369  (class class class)co 7202   ∈ cmpo 7204  Basecbs 16684  +_gcplusg 16767  +_𝑓cplusf 18083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-fv 6377  df-ov 7205  df-oprab 7206  df-mpo 7207  df-1st 7750  df-2nd 7751  df-plusf 18085

This theorem is referenced by:  plusfval  18093  plusfeq  18094  plusffn  18095  mgmplusf  18096  rlmscaf  20218  istgp2  22960  oppgtmd  22966  submtmd  22973  prdstmdd  22993  ressplusf  30927  pl1cn  31591