Theorem axcnre 7943

Description: A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 7985. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
axcnre	⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem axcnre

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

Step	Hyp	Ref	Expression
1		df-c 7880	. 2 ⊢ ℂ = (R × R)
2		eqeq1 2200	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
3	2	2rexbidv 2519	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4		opelreal 7889	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
5		opelreal 7889	. . . . . 6 ⊢ (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤 ∈ R)
6	4, 5	anbi12i 460	. . . . 5 ⊢ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ↔ (𝑧 ∈ R ∧ 𝑤 ∈ R))
7	6	biimpri 133	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ))
8		df-i 7883	. . . . . . . . 9 ⊢ i = ⟨0R, 1R⟩
9	8	oveq1i 5929	. . . . . . . 8 ⊢ (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10		0r 7812	. . . . . . . . . 10 ⊢ 0R ∈ R
11		1sr 7813	. . . . . . . . . . 11 ⊢ 1R ∈ R
12		mulcnsr 7897	. . . . . . . . . . 11 ⊢ (((0R ∈ R ∧ 1R ∈ R) ∧ (𝑤 ∈ R ∧ 0R ∈ R)) → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
13	10, 11, 12	mpanl12 436	. . . . . . . . . 10 ⊢ ((𝑤 ∈ R ∧ 0R ∈ R) → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
14	10, 13	mpan2 425	. . . . . . . . 9 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
15		mulcomsrg 7819	. . . . . . . . . . . . . 14 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R ·R 𝑤) = (𝑤 ·R 0R))
16	10, 15	mpan 424	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = (𝑤 ·R 0R))
17		00sr 7831	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 0R) = 0R)
18	16, 17	eqtrd 2226	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = 0R)
19	18	oveq1d 5934	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
20		00sr 7831	. . . . . . . . . . . . . . . 16 ⊢ (1R ∈ R → (1R ·R 0R) = 0R)
21	11, 20	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1R ·R 0R) = 0R
22	21	oveq2i 5930	. . . . . . . . . . . . . 14 ⊢ (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
23		m1r 7814	. . . . . . . . . . . . . . 15 ⊢ -1R ∈ R
24		00sr 7831	. . . . . . . . . . . . . . 15 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R)
25	23, 24	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (-1R ·R 0R) = 0R
26	22, 25	eqtri 2214	. . . . . . . . . . . . 13 ⊢ (-1R ·R (1R ·R 0R)) = 0R
27	26	oveq2i 5930	. . . . . . . . . . . 12 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
28		0idsr 7829	. . . . . . . . . . . . 13 ⊢ (0R ∈ R → (0R +R 0R) = 0R)
29	10, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0R +R 0R) = 0R
30	27, 29	eqtri 2214	. . . . . . . . . . 11 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = 0R
31	19, 30	eqtrdi 2242	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
32		mulcomsrg 7819	. . . . . . . . . . . . . 14 ⊢ ((1R ∈ R ∧ 𝑤 ∈ R) → (1R ·R 𝑤) = (𝑤 ·R 1R))
33	11, 32	mpan 424	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = (𝑤 ·R 1R))
34		1idsr 7830	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 1R) = 𝑤)
35	33, 34	eqtrd 2226	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = 𝑤)
36	35	oveq1d 5934	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
37		00sr 7831	. . . . . . . . . . . . . 14 ⊢ (0R ∈ R → (0R ·R 0R) = 0R)
38	10, 37	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0R ·R 0R) = 0R
39	38	oveq2i 5930	. . . . . . . . . . . 12 ⊢ (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
40		0idsr 7829	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (𝑤 +R 0R) = 𝑤)
41	39, 40	eqtrid 2238	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → (𝑤 +R (0R ·R 0R)) = 𝑤)
42	36, 41	eqtrd 2226	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
43	31, 42	opeq12d 3813	. . . . . . . . 9 ⊢ (𝑤 ∈ R → ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩ = ⟨0R, 𝑤⟩)
44	14, 43	eqtrd 2226	. . . . . . . 8 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
45	9, 44	eqtrid 2238	. . . . . . 7 ⊢ (𝑤 ∈ R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
46	45	oveq2d 5935	. . . . . 6 ⊢ (𝑤 ∈ R → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
47	46	adantl 277	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
48		addcnsr 7896	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 0R ∈ R) ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49	10, 48	mpanl2 435	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
50	10, 49	mpanr1 437	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
51		0idsr 7829	. . . . . 6 ⊢ (𝑧 ∈ R → (𝑧 +R 0R) = 𝑧)
52		addcomsrg 7817	. . . . . . . 8 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R +R 𝑤) = (𝑤 +R 0R))
53	10, 52	mpan 424	. . . . . . 7 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = (𝑤 +R 0R))
54	53, 40	eqtrd 2226	. . . . . 6 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = 𝑤)
55		opeq12 3807	. . . . . 6 ⊢ (((𝑧 +R 0R) = 𝑧 ∧ (0R +R 𝑤) = 𝑤) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
56	51, 54, 55	syl2an 289	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
57	47, 50, 56	3eqtrrd 2231	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
58		vex 2763	. . . . . 6 ⊢ 𝑧 ∈ V
59		opexg 4258	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 0R ∈ R) → ⟨𝑧, 0R⟩ ∈ V)
60	58, 10, 59	mp2an 426	. . . . 5 ⊢ ⟨𝑧, 0R⟩ ∈ V
61		vex 2763	. . . . . 6 ⊢ 𝑤 ∈ V
62		opexg 4258	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 0R ∈ R) → ⟨𝑤, 0R⟩ ∈ V)
63	61, 10, 62	mp2an 426	. . . . 5 ⊢ ⟨𝑤, 0R⟩ ∈ V
64		eleq1 2256	. . . . . . 7 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
65		eleq1 2256	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
66	64, 65	bi2anan9 606	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
67		oveq1 5926	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
68		oveq2 5927	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
69	68	oveq2d 5935	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
70	67, 69	sylan9eq 2246	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
71	70	eqeq2d 2205	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
72	66, 71	anbi12d 473	. . . . 5 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))) ↔ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))))
73	60, 63, 72	spc2ev 2857	. . . 4 ⊢ (((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))) → ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
74	7, 57, 73	syl2anc 411	. . 3 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
75		r2ex 2514	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
76	74, 75	sylibr 134	. 2 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)))
77	1, 3, 76	optocl 4736	1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∃wrex 2473  Vcvv 2760  ⟨cop 3622  (class class class)co 5919  Rcnr 7359  0_Rc0r 7360  1_Rc1r 7361  -1_Rcm1r 7362   +_R cplr 7363   ·_R cmr 7364  ℂcc 7872  ℝcr 7873  ici 7876   + caddc 7877   · cmul 7879

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-iplp 7530  df-imp 7531  df-enr 7788  df-nr 7789  df-plr 7790  df-mr 7791  df-0r 7793  df-1r 7794  df-m1r 7795  df-c 7880  df-i 7883  df-r 7884  df-add 7885  df-mul 7886

This theorem is referenced by: (None)