Theorem axcnre 8254

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 8296. (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 8191	. 2 ⊢ ℂ = (R × R)
2		eqeq1 2245	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
3	2	2rexbidv 2575	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4		opelreal 8200	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
5		opelreal 8200	. . . . . 6 ⊢ (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤 ∈ R)
6	4, 5	anbi12i 464	. . . . 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 8194	. . . . . . . . 9 ⊢ i = ⟨0R, 1R⟩
9	8	oveq1i 6098	. . . . . . . 8 ⊢ (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10		0r 8123	. . . . . . . . . 10 ⊢ 0R ∈ R
11		1sr 8124	. . . . . . . . . . 11 ⊢ 1R ∈ R
12		mulcnsr 8208	. . . . . . . . . . 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 440	. . . . . . . . . 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 429	. . . . . . . . 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 8130	. . . . . . . . . . . . . 14 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R ·R 𝑤) = (𝑤 ·R 0R))
16	10, 15	mpan 428	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = (𝑤 ·R 0R))
17		00sr 8142	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 0R) = 0R)
18	16, 17	eqtrd 2271	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = 0R)
19	18	oveq1d 6103	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
20		00sr 8142	. . . . . . . . . . . . . . . 16 ⊢ (1R ∈ R → (1R ·R 0R) = 0R)
21	11, 20	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1R ·R 0R) = 0R
22	21	oveq2i 6099	. . . . . . . . . . . . . 14 ⊢ (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
23		m1r 8125	. . . . . . . . . . . . . . 15 ⊢ -1R ∈ R
24		00sr 8142	. . . . . . . . . . . . . . 15 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R)
25	23, 24	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (-1R ·R 0R) = 0R
26	22, 25	eqtri 2259	. . . . . . . . . . . . 13 ⊢ (-1R ·R (1R ·R 0R)) = 0R
27	26	oveq2i 6099	. . . . . . . . . . . 12 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
28		0idsr 8140	. . . . . . . . . . . . 13 ⊢ (0R ∈ R → (0R +R 0R) = 0R)
29	10, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0R +R 0R) = 0R
30	27, 29	eqtri 2259	. . . . . . . . . . 11 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = 0R
31	19, 30	eqtrdi 2287	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
32		mulcomsrg 8130	. . . . . . . . . . . . . 14 ⊢ ((1R ∈ R ∧ 𝑤 ∈ R) → (1R ·R 𝑤) = (𝑤 ·R 1R))
33	11, 32	mpan 428	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = (𝑤 ·R 1R))
34		1idsr 8141	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 1R) = 𝑤)
35	33, 34	eqtrd 2271	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = 𝑤)
36	35	oveq1d 6103	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
37		00sr 8142	. . . . . . . . . . . . . 14 ⊢ (0R ∈ R → (0R ·R 0R) = 0R)
38	10, 37	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0R ·R 0R) = 0R
39	38	oveq2i 6099	. . . . . . . . . . . 12 ⊢ (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
40		0idsr 8140	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (𝑤 +R 0R) = 𝑤)
41	39, 40	eqtrid 2283	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → (𝑤 +R (0R ·R 0R)) = 𝑤)
42	36, 41	eqtrd 2271	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
43	31, 42	opeq12d 3912	. . . . . . . . 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 2271	. . . . . . . 8 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
45	9, 44	eqtrid 2283	. . . . . . 7 ⊢ (𝑤 ∈ R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
46	45	oveq2d 6104	. . . . . 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 8207	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 0R ∈ R) ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49	10, 48	mpanl2 439	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
50	10, 49	mpanr1 441	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
51		0idsr 8140	. . . . . 6 ⊢ (𝑧 ∈ R → (𝑧 +R 0R) = 𝑧)
52		addcomsrg 8128	. . . . . . . 8 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R +R 𝑤) = (𝑤 +R 0R))
53	10, 52	mpan 428	. . . . . . 7 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = (𝑤 +R 0R))
54	53, 40	eqtrd 2271	. . . . . 6 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = 𝑤)
55		opeq12 3906	. . . . . 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 2276	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
58		vex 2824	. . . . . 6 ⊢ 𝑧 ∈ V
59		opexg 4370	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 0R ∈ R) → ⟨𝑧, 0R⟩ ∈ V)
60	58, 10, 59	mp2an 430	. . . . 5 ⊢ ⟨𝑧, 0R⟩ ∈ V
61		vex 2824	. . . . . 6 ⊢ 𝑤 ∈ V
62		opexg 4370	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 0R ∈ R) → ⟨𝑤, 0R⟩ ∈ V)
63	61, 10, 62	mp2an 430	. . . . 5 ⊢ ⟨𝑤, 0R⟩ ∈ V
64		eleq1 2301	. . . . . . 7 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
65		eleq1 2301	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
66	64, 65	bi2anan9 614	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
67		oveq1 6095	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
68		oveq2 6096	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
69	68	oveq2d 6104	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
70	67, 69	sylan9eq 2291	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
71	70	eqeq2d 2250	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
72	66, 71	anbi12d 477	. . . . 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 2921	. . . 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 415	. . 3 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
75		r2ex 2570	. . 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 4853	1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821  ⟨cop 3712  (class class class)co 6088  Rcnr 7670  0_Rc0r 7671  1_Rc1r 7672  -1_Rcm1r 7673   +_R cplr 7674   ·_R cmr 7675  ℂcc 8183  ℝcr 8184  ici 8187   + caddc 8188   · cmul 8190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-i1p 7840  df-iplp 7841  df-imp 7842  df-enr 8099  df-nr 8100  df-plr 8101  df-mr 8102  df-0r 8104  df-1r 8105  df-m1r 8106  df-c 8191  df-i 8194  df-r 8195  df-add 8196  df-mul 8197

This theorem is used by: (None)