Theorem plusffval 17727

Description: The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

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 6496	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3		plusffval.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	2, 3	syl6eqr 2826	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5		fveq2 6496	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
6		plusffval.2	. . . . . . 7 ⊢ + = (+g‘𝐺)
7	5, 6	syl6eqr 2826	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
8	7	oveqd 6991	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥(+g‘𝑔)𝑦) = (𝑥 + 𝑦))
9	4, 4, 8	mpoeq123dv 7045	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
10		df-plusf 17721	. . . 4 ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
11		df-ov 6977	. . . . . . . 8 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
12		fvrn0 6524	. . . . . . . 8 ⊢ ( + ‘⟨𝑥, 𝑦⟩) ∈ (ran + ∪ {∅})
13	11, 12	eqeltri 2856	. . . . . . 7 ⊢ (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
14	13	rgen2w 3095	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅})
15		eqid 2772	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
16	15	fmpo 7572	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ (ran + ∪ {∅}) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)):(𝐵 × 𝐵)⟶(ran + ∪ {∅}))
17	14, 16	mpbi 222	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)):(𝐵 × 𝐵)⟶(ran + ∪ {∅})
18	3	fvexi 6510	. . . . . 6 ⊢ 𝐵 ∈ V
19	18, 18	xpex 7291	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
20	6	fvexi 6510	. . . . . . 7 ⊢ + ∈ V
21	20	rnex 7430	. . . . . 6 ⊢ ran + ∈ V
22		p0ex 5133	. . . . . 6 ⊢ {∅} ∈ V
23	21, 22	unex 7284	. . . . 5 ⊢ (ran + ∪ {∅}) ∈ V
24		fex2 7451	. . . . 5 ⊢ (((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)):(𝐵 × 𝐵)⟶(ran + ∪ {∅}) ∧ (𝐵 × 𝐵) ∈ V ∧ (ran + ∪ {∅}) ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V)
25	17, 19, 23, 24	mp3an 1440	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) ∈ V
26	9, 10, 25	fvmpt 6593	. . 3 ⊢ (𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
27		fvprc 6489	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = ∅)
28		mpo0 7053	. . . . 5 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)) = ∅
29	27, 28	syl6eqr 2826	. . . 4 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)))
30		fvprc 6489	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
31	3, 30	syl5eq 2820	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
32		mpoeq12 7043	. . . . 5 ⊢ ((𝐵 = ∅ ∧ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)))
33	31, 31, 32	syl2anc 576	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ ∅ ↦ (𝑥 + 𝑦)))
34	29, 33	eqtr4d 2811	. . 3 ⊢ (¬ 𝐺 ∈ V → (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦)))
35	26, 34	pm2.61i 177	. 2 ⊢ (+𝑓‘𝐺) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
36	1, 35	eqtri 2796	1 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))

Syntax hints:  ¬ wn 3   = wceq 1507   ∈ wcel 2050  ∀wral 3082  Vcvv 3409   ∪ cun 3821  ∅c0 4172  {csn 4435  ⟨cop 4441   × cxp 5401  ran crn 5404  ⟶wf 6181  ‘cfv 6185  (class class class)co 6974   ∈ cmpo 6976  Basecbs 16337  +_gcplusg 16419  +_𝑓cplusf 17719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-plusf 17721

This theorem is referenced by:  plusfval  17728  plusfeq  17729  plusffn  17730  mgmplusf  17731  rlmscaf  19714  istgp2  22415  oppgtmd  22421  submtmd  22428  prdstmdd  22447  ressplusf  30392  pl1cn  30871