Theorem axaddcl 8178

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 8222 be used later. Instead, in most cases use addcl 8251. (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 4764	. . . . 5 ⊢ (𝐴 ∈ (R × R) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
2		df-c 8132	. . . . 5 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2327	. . . 4 ⊢ (𝐴 ∈ ℂ → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
4		elxpi 4764	. . . . 5 ⊢ (𝐵 ∈ (R × R) → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
5	4, 2	eleq2s 2327	. . . 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 1988	. . 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 6067	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))
12		addcnsr 8148	. . . . . . 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 2265	. . . . 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 8067	. . . . . . . . 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 8067	. . . . . . . . 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 4780	. . . . . . . 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 2326	. . . . . 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 2309	. . . 4 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
24	23	exlimivv 1946	. . 3 ⊢ (∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 + 𝐵) ∈ ℂ)
25	24	exlimivv 1946	. 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 2203  ⟨cop 3691   × cxp 4746  (class class class)co 6049  Rcnr 7611   +_R cplr 7615  ℂcc 8124   + caddc 8129

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-enq0 7738  df-nq0 7739  df-0nq0 7740  df-plq0 7741  df-mq0 7742  df-inp 7780  df-iplp 7782  df-enr 8040  df-nr 8041  df-plr 8042  df-c 8132  df-add 8137

This theorem is referenced by:  axaddf  8182