Theorem axcnre 7843

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 7885. (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 7780	. 2 ⊢ ℂ = (R × R)
2		eqeq1 2177	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
3	2	2rexbidv 2495	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4		opelreal 7789	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
5		opelreal 7789	. . . . . 6 ⊢ (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤 ∈ R)
6	4, 5	anbi12i 457	. . . . 5 ⊢ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ↔ (𝑧 ∈ R ∧ 𝑤 ∈ R))
7	6	biimpri 132	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ))
8		df-i 7783	. . . . . . . . 9 ⊢ i = ⟨0R, 1R⟩
9	8	oveq1i 5863	. . . . . . . 8 ⊢ (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10		0r 7712	. . . . . . . . . 10 ⊢ 0R ∈ R
11		1sr 7713	. . . . . . . . . . 11 ⊢ 1R ∈ R
12		mulcnsr 7797	. . . . . . . . . . 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 434	. . . . . . . . . 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 423	. . . . . . . . 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 7719	. . . . . . . . . . . . . 14 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R ·R 𝑤) = (𝑤 ·R 0R))
16	10, 15	mpan 422	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = (𝑤 ·R 0R))
17		00sr 7731	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 0R) = 0R)
18	16, 17	eqtrd 2203	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = 0R)
19	18	oveq1d 5868	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
20		00sr 7731	. . . . . . . . . . . . . . . 16 ⊢ (1R ∈ R → (1R ·R 0R) = 0R)
21	11, 20	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1R ·R 0R) = 0R
22	21	oveq2i 5864	. . . . . . . . . . . . . 14 ⊢ (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
23		m1r 7714	. . . . . . . . . . . . . . 15 ⊢ -1R ∈ R
24		00sr 7731	. . . . . . . . . . . . . . 15 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R)
25	23, 24	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (-1R ·R 0R) = 0R
26	22, 25	eqtri 2191	. . . . . . . . . . . . 13 ⊢ (-1R ·R (1R ·R 0R)) = 0R
27	26	oveq2i 5864	. . . . . . . . . . . 12 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
28		0idsr 7729	. . . . . . . . . . . . 13 ⊢ (0R ∈ R → (0R +R 0R) = 0R)
29	10, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0R +R 0R) = 0R
30	27, 29	eqtri 2191	. . . . . . . . . . 11 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = 0R
31	19, 30	eqtrdi 2219	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
32		mulcomsrg 7719	. . . . . . . . . . . . . 14 ⊢ ((1R ∈ R ∧ 𝑤 ∈ R) → (1R ·R 𝑤) = (𝑤 ·R 1R))
33	11, 32	mpan 422	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = (𝑤 ·R 1R))
34		1idsr 7730	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 1R) = 𝑤)
35	33, 34	eqtrd 2203	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = 𝑤)
36	35	oveq1d 5868	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
37		00sr 7731	. . . . . . . . . . . . . 14 ⊢ (0R ∈ R → (0R ·R 0R) = 0R)
38	10, 37	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0R ·R 0R) = 0R
39	38	oveq2i 5864	. . . . . . . . . . . 12 ⊢ (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
40		0idsr 7729	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (𝑤 +R 0R) = 𝑤)
41	39, 40	eqtrid 2215	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → (𝑤 +R (0R ·R 0R)) = 𝑤)
42	36, 41	eqtrd 2203	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
43	31, 42	opeq12d 3773	. . . . . . . . 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 2203	. . . . . . . 8 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
45	9, 44	eqtrid 2215	. . . . . . 7 ⊢ (𝑤 ∈ R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
46	45	oveq2d 5869	. . . . . 6 ⊢ (𝑤 ∈ R → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
47	46	adantl 275	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
48		addcnsr 7796	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 0R ∈ R) ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49	10, 48	mpanl2 433	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
50	10, 49	mpanr1 435	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
51		0idsr 7729	. . . . . 6 ⊢ (𝑧 ∈ R → (𝑧 +R 0R) = 𝑧)
52		addcomsrg 7717	. . . . . . . 8 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R +R 𝑤) = (𝑤 +R 0R))
53	10, 52	mpan 422	. . . . . . 7 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = (𝑤 +R 0R))
54	53, 40	eqtrd 2203	. . . . . 6 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = 𝑤)
55		opeq12 3767	. . . . . 6 ⊢ (((𝑧 +R 0R) = 𝑧 ∧ (0R +R 𝑤) = 𝑤) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
56	51, 54, 55	syl2an 287	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
57	47, 50, 56	3eqtrrd 2208	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
58		vex 2733	. . . . . 6 ⊢ 𝑧 ∈ V
59		opexg 4213	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 0R ∈ R) → ⟨𝑧, 0R⟩ ∈ V)
60	58, 10, 59	mp2an 424	. . . . 5 ⊢ ⟨𝑧, 0R⟩ ∈ V
61		vex 2733	. . . . . 6 ⊢ 𝑤 ∈ V
62		opexg 4213	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 0R ∈ R) → ⟨𝑤, 0R⟩ ∈ V)
63	61, 10, 62	mp2an 424	. . . . 5 ⊢ ⟨𝑤, 0R⟩ ∈ V
64		eleq1 2233	. . . . . . 7 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
65		eleq1 2233	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
66	64, 65	bi2anan9 601	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
67		oveq1 5860	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
68		oveq2 5861	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
69	68	oveq2d 5869	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
70	67, 69	sylan9eq 2223	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
71	70	eqeq2d 2182	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
72	66, 71	anbi12d 470	. . . . 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 2826	. . . 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 409	. . 3 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥∃𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
75		r2ex 2490	. . 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 133	. 2 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)))
77	1, 3, 76	optocl 4687	1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449  Vcvv 2730  ⟨cop 3586  (class class class)co 5853  Rcnr 7259  0_Rc0r 7260  1_Rc1r 7261  -1_Rcm1r 7262   +_R cplr 7263   ·_R cmr 7264  ℂcc 7772  ℝcr 7773  ici 7776   + caddc 7777   · cmul 7779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-imp 7431  df-enr 7688  df-nr 7689  df-plr 7690  df-mr 7691  df-0r 7693  df-1r 7694  df-m1r 7695  df-c 7780  df-i 7783  df-r 7784  df-add 7785  df-mul 7786

This theorem is referenced by: (None)