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Theorem axcnre 10023
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 10047. (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.
StepHypRef Expression
1 df-c 9980 . 2 ℂ = (R × R)
2 eqeq1 2655 . . 3 (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
322rexbidv 3086 . 2 (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4 opelreal 9989 . . . . . 6 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
5 opelreal 9989 . . . . . 6 (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤R)
64, 5anbi12i 733 . . . . 5 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ↔ (𝑧R𝑤R))
76biimpri 218 . . . 4 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ))
8 df-i 9983 . . . . . . . . 9 i = ⟨0R, 1R
98oveq1i 6700 . . . . . . . 8 (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10 0r 9939 . . . . . . . . . 10 0RR
11 1sr 9940 . . . . . . . . . . 11 1RR
12 mulcnsr 9995 . . . . . . . . . . 11 (((0RR ∧ 1RR) ∧ (𝑤R ∧ 0RR)) → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
1310, 11, 12mpanl12 718 . . . . . . . . . 10 ((𝑤R ∧ 0RR) → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
1410, 13mpan2 707 . . . . . . . . 9 (𝑤R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
15 mulcomsr 9948 . . . . . . . . . . . . 13 (0R ·R 𝑤) = (𝑤 ·R 0R)
16 00sr 9958 . . . . . . . . . . . . 13 (𝑤R → (𝑤 ·R 0R) = 0R)
1715, 16syl5eq 2697 . . . . . . . . . . . 12 (𝑤R → (0R ·R 𝑤) = 0R)
1817oveq1d 6705 . . . . . . . . . . 11 (𝑤R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
19 00sr 9958 . . . . . . . . . . . . . . . 16 (1RR → (1R ·R 0R) = 0R)
2011, 19ax-mp 5 . . . . . . . . . . . . . . 15 (1R ·R 0R) = 0R
2120oveq2i 6701 . . . . . . . . . . . . . 14 (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
22 m1r 9941 . . . . . . . . . . . . . . 15 -1RR
23 00sr 9958 . . . . . . . . . . . . . . 15 (-1RR → (-1R ·R 0R) = 0R)
2422, 23ax-mp 5 . . . . . . . . . . . . . 14 (-1R ·R 0R) = 0R
2521, 24eqtri 2673 . . . . . . . . . . . . 13 (-1R ·R (1R ·R 0R)) = 0R
2625oveq2i 6701 . . . . . . . . . . . 12 (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
27 0idsr 9956 . . . . . . . . . . . . 13 (0RR → (0R +R 0R) = 0R)
2810, 27ax-mp 5 . . . . . . . . . . . 12 (0R +R 0R) = 0R
2926, 28eqtri 2673 . . . . . . . . . . 11 (0R +R (-1R ·R (1R ·R 0R))) = 0R
3018, 29syl6eq 2701 . . . . . . . . . 10 (𝑤R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
31 mulcomsr 9948 . . . . . . . . . . . . 13 (1R ·R 𝑤) = (𝑤 ·R 1R)
32 1idsr 9957 . . . . . . . . . . . . 13 (𝑤R → (𝑤 ·R 1R) = 𝑤)
3331, 32syl5eq 2697 . . . . . . . . . . . 12 (𝑤R → (1R ·R 𝑤) = 𝑤)
3433oveq1d 6705 . . . . . . . . . . 11 (𝑤R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
35 00sr 9958 . . . . . . . . . . . . . 14 (0RR → (0R ·R 0R) = 0R)
3610, 35ax-mp 5 . . . . . . . . . . . . 13 (0R ·R 0R) = 0R
3736oveq2i 6701 . . . . . . . . . . . 12 (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
38 0idsr 9956 . . . . . . . . . . . 12 (𝑤R → (𝑤 +R 0R) = 𝑤)
3937, 38syl5eq 2697 . . . . . . . . . . 11 (𝑤R → (𝑤 +R (0R ·R 0R)) = 𝑤)
4034, 39eqtrd 2685 . . . . . . . . . 10 (𝑤R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
4130, 40opeq12d 4441 . . . . . . . . 9 (𝑤R → ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩ = ⟨0R, 𝑤⟩)
4214, 41eqtrd 2685 . . . . . . . 8 (𝑤R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
439, 42syl5eq 2697 . . . . . . 7 (𝑤R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
4443oveq2d 6706 . . . . . 6 (𝑤R → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
4544adantl 481 . . . . 5 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
46 addcnsr 9994 . . . . . . 7 (((𝑧R ∧ 0RR) ∧ (0RR𝑤R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
4710, 46mpanl2 717 . . . . . 6 ((𝑧R ∧ (0RR𝑤R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
4810, 47mpanr1 719 . . . . 5 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49 0idsr 9956 . . . . . 6 (𝑧R → (𝑧 +R 0R) = 𝑧)
50 addcomsr 9946 . . . . . . 7 (0R +R 𝑤) = (𝑤 +R 0R)
5150, 38syl5eq 2697 . . . . . 6 (𝑤R → (0R +R 𝑤) = 𝑤)
52 opeq12 4435 . . . . . 6 (((𝑧 +R 0R) = 𝑧 ∧ (0R +R 𝑤) = 𝑤) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
5349, 51, 52syl2an 493 . . . . 5 ((𝑧R𝑤R) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
5445, 48, 533eqtrrd 2690 . . . 4 ((𝑧R𝑤R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
55 opex 4962 . . . . 5 𝑧, 0R⟩ ∈ V
56 opex 4962 . . . . 5 𝑤, 0R⟩ ∈ V
57 eleq1 2718 . . . . . . 7 (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
58 eleq1 2718 . . . . . . 7 (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
5957, 58bi2anan9 935 . . . . . 6 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
60 oveq1 6697 . . . . . . . 8 (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
61 oveq2 6698 . . . . . . . . 9 (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
6261oveq2d 6706 . . . . . . . 8 (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
6360, 62sylan9eq 2705 . . . . . . 7 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
6463eqeq2d 2661 . . . . . 6 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
6559, 64anbi12d 747 . . . . 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⟩)))))
6655, 56, 65spc2ev 3332 . . . 4 (((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))) → ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
677, 54, 66syl2anc 694 . . 3 ((𝑧R𝑤R) → ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
68 r2ex 3090 . . 3 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
6967, 68sylibr 224 . 2 ((𝑧R𝑤R) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)))
701, 3, 69optocl 5229 1 (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  cop 4216  (class class class)co 6690  Rcnr 9725  0Rc0r 9726  1Rc1r 9727  -1Rcm1r 9728   +R cplr 9729   ·R cmr 9730  cc 9972  cr 9973  ici 9976   + caddc 9977   · cmul 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-mr 9918  df-0r 9920  df-1r 9921  df-m1r 9922  df-c 9980  df-i 9983  df-r 9984  df-add 9985  df-mul 9986
This theorem is referenced by: (None)
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