Theorem axcnre 7822

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 7864. (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 7759	. 2 ⊢ ℂ = (R × R)
2		eqeq1 2172	. . 3 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
3	2	2rexbidv 2491	. 2 ⊢ (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4		opelreal 7768	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
5		opelreal 7768	. . . . . 6 ⊢ (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤 ∈ R)
6	4, 5	anbi12i 456	. . . . 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 7762	. . . . . . . . 9 ⊢ i = ⟨0R, 1R⟩
9	8	oveq1i 5852	. . . . . . . 8 ⊢ (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10		0r 7691	. . . . . . . . . 10 ⊢ 0R ∈ R
11		1sr 7692	. . . . . . . . . . 11 ⊢ 1R ∈ R
12		mulcnsr 7776	. . . . . . . . . . 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 433	. . . . . . . . . 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 422	. . . . . . . . 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 7698	. . . . . . . . . . . . . 14 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R ·R 𝑤) = (𝑤 ·R 0R))
16	10, 15	mpan 421	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = (𝑤 ·R 0R))
17		00sr 7710	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 0R) = 0R)
18	16, 17	eqtrd 2198	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (0R ·R 𝑤) = 0R)
19	18	oveq1d 5857	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
20		00sr 7710	. . . . . . . . . . . . . . . 16 ⊢ (1R ∈ R → (1R ·R 0R) = 0R)
21	11, 20	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1R ·R 0R) = 0R
22	21	oveq2i 5853	. . . . . . . . . . . . . 14 ⊢ (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
23		m1r 7693	. . . . . . . . . . . . . . 15 ⊢ -1R ∈ R
24		00sr 7710	. . . . . . . . . . . . . . 15 ⊢ (-1R ∈ R → (-1R ·R 0R) = 0R)
25	23, 24	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (-1R ·R 0R) = 0R
26	22, 25	eqtri 2186	. . . . . . . . . . . . 13 ⊢ (-1R ·R (1R ·R 0R)) = 0R
27	26	oveq2i 5853	. . . . . . . . . . . 12 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
28		0idsr 7708	. . . . . . . . . . . . 13 ⊢ (0R ∈ R → (0R +R 0R) = 0R)
29	10, 28	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0R +R 0R) = 0R
30	27, 29	eqtri 2186	. . . . . . . . . . 11 ⊢ (0R +R (-1R ·R (1R ·R 0R))) = 0R
31	19, 30	eqtrdi 2215	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
32		mulcomsrg 7698	. . . . . . . . . . . . . 14 ⊢ ((1R ∈ R ∧ 𝑤 ∈ R) → (1R ·R 𝑤) = (𝑤 ·R 1R))
33	11, 32	mpan 421	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = (𝑤 ·R 1R))
34		1idsr 7709	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ R → (𝑤 ·R 1R) = 𝑤)
35	33, 34	eqtrd 2198	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (1R ·R 𝑤) = 𝑤)
36	35	oveq1d 5857	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
37		00sr 7710	. . . . . . . . . . . . . 14 ⊢ (0R ∈ R → (0R ·R 0R) = 0R)
38	10, 37	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0R ·R 0R) = 0R
39	38	oveq2i 5853	. . . . . . . . . . . 12 ⊢ (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
40		0idsr 7708	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ R → (𝑤 +R 0R) = 𝑤)
41	39, 40	syl5eq 2211	. . . . . . . . . . 11 ⊢ (𝑤 ∈ R → (𝑤 +R (0R ·R 0R)) = 𝑤)
42	36, 41	eqtrd 2198	. . . . . . . . . 10 ⊢ (𝑤 ∈ R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
43	31, 42	opeq12d 3766	. . . . . . . . 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 2198	. . . . . . . 8 ⊢ (𝑤 ∈ R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
45	9, 44	syl5eq 2211	. . . . . . 7 ⊢ (𝑤 ∈ R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
46	45	oveq2d 5858	. . . . . 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 7775	. . . . . . 7 ⊢ (((𝑧 ∈ R ∧ 0R ∈ R) ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49	10, 48	mpanl2 432	. . . . . 6 ⊢ ((𝑧 ∈ R ∧ (0R ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
50	10, 49	mpanr1 434	. . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
51		0idsr 7708	. . . . . 6 ⊢ (𝑧 ∈ R → (𝑧 +R 0R) = 𝑧)
52		addcomsrg 7696	. . . . . . . 8 ⊢ ((0R ∈ R ∧ 𝑤 ∈ R) → (0R +R 𝑤) = (𝑤 +R 0R))
53	10, 52	mpan 421	. . . . . . 7 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = (𝑤 +R 0R))
54	53, 40	eqtrd 2198	. . . . . 6 ⊢ (𝑤 ∈ R → (0R +R 𝑤) = 𝑤)
55		opeq12 3760	. . . . . 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 2203	. . . 4 ⊢ ((𝑧 ∈ R ∧ 𝑤 ∈ R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
58		vex 2729	. . . . . 6 ⊢ 𝑧 ∈ V
59		opexg 4206	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 0R ∈ R) → ⟨𝑧, 0R⟩ ∈ V)
60	58, 10, 59	mp2an 423	. . . . 5 ⊢ ⟨𝑧, 0R⟩ ∈ V
61		vex 2729	. . . . . 6 ⊢ 𝑤 ∈ V
62		opexg 4206	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 0R ∈ R) → ⟨𝑤, 0R⟩ ∈ V)
63	61, 10, 62	mp2an 423	. . . . 5 ⊢ ⟨𝑤, 0R⟩ ∈ V
64		eleq1 2229	. . . . . . 7 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
65		eleq1 2229	. . . . . . 7 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
66	64, 65	bi2anan9 596	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
67		oveq1 5849	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
68		oveq2 5850	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
69	68	oveq2d 5858	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
70	67, 69	sylan9eq 2219	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
71	70	eqeq2d 2177	. . . . . 6 ⊢ ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
72	66, 71	anbi12d 465	. . . . 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 2822	. . . 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 2486	. . 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 4680	1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  Vcvv 2726  ⟨cop 3579  (class class class)co 5842  Rcnr 7238  0_Rc0r 7239  1_Rc1r 7240  -1_Rcm1r 7241   +_R cplr 7242   ·_R cmr 7243  ℂcc 7751  ℝcr 7752  ici 7755   + caddc 7756   · cmul 7758

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-imp 7410  df-enr 7667  df-nr 7668  df-plr 7669  df-mr 7670  df-0r 7672  df-1r 7673  df-m1r 7674  df-c 7759  df-i 7762  df-r 7763  df-add 7764  df-mul 7765

This theorem is referenced by: (None)