Theorem axcnre 10851

Description: A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 10875. (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 10808	. 2 ⊢ ℂ = (R × R)
2		eqeq1 2742	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
3	2	2rexbidv 3228	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4		opelreal 10817	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
5		opelreal 10817	. . . . . 6 ⊢ (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤 ∈ R)
6	4, 5	anbi12i 626	. . . . 5 ⊢ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ↔ (𝑧 ∈ R ∧ 𝑤 ∈ R))
7	6	biimpri 227	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ))
8		df-i 10811	. . . . . . . . 9 ⊢ i = ⟨0R, 1R⟩
9	8	oveq1i 7265	. . . . . . . 8 ⊢ (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10		0r 10767	. . . . . . . . . 10 ⊢ 0R ∈ R
11		1sr 10768	. . . . . . . . . . 11 ⊢ 1R ∈ R
12		mulcnsr 10823	. . . . . . . . . . 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 698	. . . . . . . . . 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 687	. . . . . . . . 9 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
15		mulcomsr 10776	. . . . . . . . . . . . 13 ⊢ (0R ·R 𝑤) = (𝑤 ·R 0R)
16		00sr 10786	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 0R) = 0R)
17	15, 16	eqtrid 2790	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = 0R)
18	17	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
19		00sr 10786	. . . . . . . . . . . . . . . 16 ⊢ (1R ∈ R → (1R ·R 0R) = 0R)
20	11, 19	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1R ·R 0R) = 0R
21	20	oveq2i 7266	. . . . . . . . . . . . . 14 ⊢ (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
22		m1r 10769	. . . . . . . . . . . . . . 15 ⊢ -1R ∈ R
23		00sr 10786	. . . . . . . . . . . . . . 15 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (-1R ·R 0R) = 0R
25	21, 24	eqtri 2766	. . . . . . . . . . . . 13 ⊢ (-1R ·R (1R ·R 0R)) = 0R
26	25	oveq2i 7266	. . . . . . . . . . . 12 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
27		0idsr 10784	. . . . . . . . . . . . 13 ⊢ (0R ∈ R → (0R +R 0R) = 0R)
28	10, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0R +R 0R) = 0R
29	26, 28	eqtri 2766	. . . . . . . . . . 11 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = 0R
30	18, 29	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
31		mulcomsr 10776	. . . . . . . . . . . . 13 ⊢ (1R ·R 𝑤) = (𝑤 ·R 1R)
32		1idsr 10785	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 1R) = 𝑤)
33	31, 32	eqtrid 2790	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = 𝑤)
34	33	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
35		00sr 10786	. . . . . . . . . . . . . 14 ⊢ (0R ∈ R → (0R ·R 0R) = 0R)
36	10, 35	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0R ·R 0R) = 0R
37	36	oveq2i 7266	. . . . . . . . . . . 12 ⊢ (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
38		0idsr 10784	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (𝑤 +R 0R) = 𝑤)
39	37, 38	eqtrid 2790	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → (𝑤 +R (0R ·R 0R)) = 𝑤)
40	34, 39	eqtrd 2778	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
41	30, 40	opeq12d 4809	. . . . . . . . 9 ⊢ (𝑤 ∈ R → ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩ = ⟨0R, 𝑤⟩)
42	14, 41	eqtrd 2778	. . . . . . . 8 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
43	9, 42	eqtrid 2790	. . . . . . 7 ⊢ (𝑤 ∈ R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
44	43	oveq2d 7271	. . . . . 6 ⊢ (𝑤 ∈ R → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
45	44	adantl 481	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
46		addcnsr 10822	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 0R ∈ R) ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
47	10, 46	mpanl2 697	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
48	10, 47	mpanr1 699	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49		0idsr 10784	. . . . . 6 ⊢ (𝑧 ∈ R → (𝑧 +R 0R) = 𝑧)
50		addcomsr 10774	. . . . . . 7 ⊢ (0R +R 𝑤) = (𝑤 +R 0R)
51	50, 38	eqtrid 2790	. . . . . 6 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = 𝑤)
52		opeq12 4803	. . . . . 6 ⊢ (((𝑧 +R 0R) = 𝑧 ∧ (0R +R 𝑤) = 𝑤) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
53	49, 51, 52	syl2an 595	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
54	45, 48, 53	3eqtrrd 2783	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
55		opex 5373	. . . . 5 ⊢ ⟨𝑧, 0R⟩ ∈ V
56		opex 5373	. . . . 5 ⊢ ⟨𝑤, 0R⟩ ∈ V
57		eleq1 2826	. . . . . . 7 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
58		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
59	57, 58	bi2anan9 635	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
60		oveq1 7262	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
61		oveq2 7263	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
62	61	oveq2d 7271	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
63	60, 62	sylan9eq 2799	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
64	63	eqeq2d 2749	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
65	59, 64	anbi12d 630	. . . . 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⟩)))))
66	55, 56, 65	spc2ev 3536	. . . 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))))
67	7, 54, 66	syl2anc 583	. . 3 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
68		r2ex 3231	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
69	67, 68	sylibr 233	. 2 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)))
70	1, 3, 69	optocl 5671	1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064  ⟨cop 4564  (class class class)co 7255  Rcnr 10552  0_Rc0r 10553  1_Rc1r 10554  -1_Rcm1r 10555   +_R cplr 10556   ·_R cmr 10557  ℂcc 10800  ℝcr 10801  ici 10804   + caddc 10805   · cmul 10807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-ec 8458  df-qs 8462  df-ni 10559  df-pli 10560  df-mi 10561  df-lti 10562  df-plpq 10595  df-mpq 10596  df-ltpq 10597  df-enq 10598  df-nq 10599  df-erq 10600  df-plq 10601  df-mq 10602  df-1nq 10603  df-rq 10604  df-ltnq 10605  df-np 10668  df-1p 10669  df-plp 10670  df-mp 10671  df-ltp 10672  df-enr 10742  df-nr 10743  df-plr 10744  df-mr 10745  df-0r 10747  df-1r 10748  df-m1r 10749  df-c 10808  df-i 10811  df-r 10812  df-add 10813  df-mul 10814

This theorem is referenced by: (None)