Theorem axaddcl 8127

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 8171 be used later. Instead, in most cases use addcl 8200. (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 4747	. . . . 5 ⊢ (𝐴 ∈ (R × R) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
2		df-c 8081	. . . . 5 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2326	. . . 4 ⊢ (𝐴 ∈ ℂ → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
4		elxpi 4747	. . . . 5 ⊢ (𝐵 ∈ (R × R) → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
5	4, 2	eleq2s 2326	. . . 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 1987	. . 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 531	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐵 = ⟨𝑧, 𝑤⟩)
11	9, 10	oveq12d 6046	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))
12		addcnsr 8097	. . . . . . 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 2264	. . . . 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 8016	. . . . . . . . 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 8016	. . . . . . . . 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 4763	. . . . . . . 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 2325	. . . . . 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 2308	. . . 4 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
24	23	exlimivv 1945	. . 3 ⊢ (∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
25	24	exlimivv 1945	. 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 1398  ∃wex 1541   ∈ wcel 2202  ⟨cop 3676   × cxp 4729  (class class class)co 6028  Rcnr 7560   +_R cplr 7564  ℂcc 8073   + caddc 8078

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-enr 7989  df-nr 7990  df-plr 7991  df-c 8081  df-add 8086

This theorem is referenced by:  axaddf  8131