Theorem axaddcl 8077

Description: Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 8121 be used later. Instead, in most cases use addcl 8150. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
axaddcl	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

Proof of Theorem axaddcl

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxpi 4739	. . . . 5 ⊢ (𝐴 ∈ (R × R) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
2		df-c 8031	. . . . 5 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2324	. . . 4 ⊢ (𝐴 ∈ ℂ → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
4		elxpi 4739	. . . . 5 ⊢ (𝐵 ∈ (R × R) → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
5	4, 2	eleq2s 2324	. . . 4 ⊢ (𝐵 ∈ ℂ → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
6	3, 5	anim12i 338	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))))
7		ee4anv 1985	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) ↔ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))))
8	6, 7	sylibr 134	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑥∃𝑦∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))))
9		simpll 527	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐴 = ⟨𝑥, 𝑦⟩)
10		simprl 529	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐵 = ⟨𝑧, 𝑤⟩)
11	9, 10	oveq12d 6031	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))
12		addcnsr 8047	. . . . . . 7 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
13	12	ad2ant2l 508	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
14	11, 13	eqtrd 2262	. . . . 5 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) = ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩)
15		addclsr 7966	. . . . . . . . 9 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 +R 𝑧) ∈ R)
16	15	ad2ant2r 509	. . . . . . . 8 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑥 +R 𝑧) ∈ R)
17		addclsr 7966	. . . . . . . . 9 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 +R 𝑤) ∈ R)
18	17	ad2ant2l 508	. . . . . . . 8 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑦 +R 𝑤) ∈ R)
19		opelxpi 4755	. . . . . . . 8 ⊢ (((𝑥 +R 𝑧) ∈ R ∧ (𝑦 +R 𝑤) ∈ R) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ ∈ (R × R))
20	16, 18, 19	syl2anc 411	. . . . . . 7 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ ∈ (R × R))
21	20, 2	eleqtrrdi 2323	. . . . . 6 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ ∈ ℂ)
22	21	ad2ant2l 508	. . . . 5 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → ⟨(𝑥 +R 𝑧), (𝑦 +R 𝑤)⟩ ∈ ℂ)
23	14, 22	eqeltrd 2306	. . . 4 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
24	23	exlimivv 1943	. . 3 ⊢ (∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
25	24	exlimivv 1943	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
26	8, 25	syl 14	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ⟨cop 3670   × cxp 4721  (class class class)co 6013  Rcnr 7510   +_R cplr 7514  ℂcc 8023   + caddc 8028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-enq0 7637  df-nq0 7638  df-0nq0 7639  df-plq0 7640  df-mq0 7641  df-inp 7679  df-iplp 7681  df-enr 7939  df-nr 7940  df-plr 7941  df-c 8031  df-add 8036

This theorem is referenced by:  axaddf  8081