Theorem plusffval 18663

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 6863	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		plusffval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6863	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		plusffval.2	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	eqtr4di 2814	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 7409	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
9	4, 4, 8	mpoeq123dv 7467	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
10		df-plusf 18656	. . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
11	3	fvexi 6877	. . . . 5 ⊢ 𝐵 ∈ V
12	6	fvexi 6877	. . . . . . 7 ⊢ + ∈ V
13	12	rnex 7887	. . . . . 6 ⊢ ran + ∈ V
14		p0ex 5340	. . . . . 6 ⊢ {∅} ∈ V
15	13, 14	unex 7723	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
16		df-ov 7395	. . . . . . 7 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
17		fvrn0 6891	. . . . . . 7 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
18	16, 17	eqeltri 2857	. . . . . 6 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
19	18	rgen2w 3080	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
20	11, 11, 15, 19	mpoexw 8055	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V
21	9, 10, 20	fvmpt 6971	. . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
22		fvprc 6855	. . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅)
23		fvprc 6855	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
24	3, 23	eqtrid 2808	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
25	24	olcd 885	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
26		0mpo0 7475	. . . . 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 2799	. . 3 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
29	21, 28	pm2.61i 183	. 2 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
30	1, 29	eqtri 2784	1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))

Syntax hints:  ¬ wn 3   ∨ wo 858   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∪ cun 3902  ∅c0 4285  {csn 4581  ⟨cop 4587  ran crn 5646  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Basecbs 17228  +_gcplusg 17269  +_𝑓cplusf 18654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-plusf 18656

This theorem is referenced by:  plusfval  18664  plusfeq  18665  plusffn  18666  mgmplusf  18667  rlmscaf  21254  istgp2  24131  oppgtmd  24137  submtmd  24144  prdstmdd  24164  ressplusf  33102  pl1cn  34213