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Theorem axcnre 10579
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 10603. (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 10536 . 2 ℂ = (R × R)
2 eqeq1 2805 . . 3 (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
322rexbidv 3262 . 2 (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4 opelreal 10545 . . . . . 6 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
5 opelreal 10545 . . . . . 6 (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤R)
64, 5anbi12i 629 . . . . 5 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ↔ (𝑧R𝑤R))
76biimpri 231 . . . 4 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ))
8 df-i 10539 . . . . . . . . 9 i = ⟨0R, 1R
98oveq1i 7149 . . . . . . . 8 (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10 0r 10495 . . . . . . . . . 10 0RR
11 1sr 10496 . . . . . . . . . . 11 1RR
12 mulcnsr 10551 . . . . . . . . . . 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 701 . . . . . . . . . 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 690 . . . . . . . . 9 (𝑤R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
15 mulcomsr 10504 . . . . . . . . . . . . 13 (0R ·R 𝑤) = (𝑤 ·R 0R)
16 00sr 10514 . . . . . . . . . . . . 13 (𝑤R → (𝑤 ·R 0R) = 0R)
1715, 16syl5eq 2848 . . . . . . . . . . . 12 (𝑤R → (0R ·R 𝑤) = 0R)
1817oveq1d 7154 . . . . . . . . . . 11 (𝑤R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
19 00sr 10514 . . . . . . . . . . . . . . . 16 (1RR → (1R ·R 0R) = 0R)
2011, 19ax-mp 5 . . . . . . . . . . . . . . 15 (1R ·R 0R) = 0R
2120oveq2i 7150 . . . . . . . . . . . . . 14 (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
22 m1r 10497 . . . . . . . . . . . . . . 15 -1RR
23 00sr 10514 . . . . . . . . . . . . . . 15 (-1RR → (-1R ·R 0R) = 0R)
2422, 23ax-mp 5 . . . . . . . . . . . . . 14 (-1R ·R 0R) = 0R
2521, 24eqtri 2824 . . . . . . . . . . . . 13 (-1R ·R (1R ·R 0R)) = 0R
2625oveq2i 7150 . . . . . . . . . . . 12 (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
27 0idsr 10512 . . . . . . . . . . . . 13 (0RR → (0R +R 0R) = 0R)
2810, 27ax-mp 5 . . . . . . . . . . . 12 (0R +R 0R) = 0R
2926, 28eqtri 2824 . . . . . . . . . . 11 (0R +R (-1R ·R (1R ·R 0R))) = 0R
3018, 29eqtrdi 2852 . . . . . . . . . 10 (𝑤R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
31 mulcomsr 10504 . . . . . . . . . . . . 13 (1R ·R 𝑤) = (𝑤 ·R 1R)
32 1idsr 10513 . . . . . . . . . . . . 13 (𝑤R → (𝑤 ·R 1R) = 𝑤)
3331, 32syl5eq 2848 . . . . . . . . . . . 12 (𝑤R → (1R ·R 𝑤) = 𝑤)
3433oveq1d 7154 . . . . . . . . . . 11 (𝑤R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
35 00sr 10514 . . . . . . . . . . . . . 14 (0RR → (0R ·R 0R) = 0R)
3610, 35ax-mp 5 . . . . . . . . . . . . 13 (0R ·R 0R) = 0R
3736oveq2i 7150 . . . . . . . . . . . 12 (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
38 0idsr 10512 . . . . . . . . . . . 12 (𝑤R → (𝑤 +R 0R) = 𝑤)
3937, 38syl5eq 2848 . . . . . . . . . . 11 (𝑤R → (𝑤 +R (0R ·R 0R)) = 𝑤)
4034, 39eqtrd 2836 . . . . . . . . . 10 (𝑤R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
4130, 40opeq12d 4776 . . . . . . . . 9 (𝑤R → ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩ = ⟨0R, 𝑤⟩)
4214, 41eqtrd 2836 . . . . . . . 8 (𝑤R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
439, 42syl5eq 2848 . . . . . . 7 (𝑤R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
4443oveq2d 7155 . . . . . 6 (𝑤R → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
4544adantl 485 . . . . 5 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)) = (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩))
46 addcnsr 10550 . . . . . . 7 (((𝑧R ∧ 0RR) ∧ (0RR𝑤R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
4710, 46mpanl2 700 . . . . . 6 ((𝑧R ∧ (0RR𝑤R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
4810, 47mpanr1 702 . . . . 5 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49 0idsr 10512 . . . . . 6 (𝑧R → (𝑧 +R 0R) = 𝑧)
50 addcomsr 10502 . . . . . . 7 (0R +R 𝑤) = (𝑤 +R 0R)
5150, 38syl5eq 2848 . . . . . 6 (𝑤R → (0R +R 𝑤) = 𝑤)
52 opeq12 4770 . . . . . 6 (((𝑧 +R 0R) = 𝑧 ∧ (0R +R 𝑤) = 𝑤) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
5349, 51, 52syl2an 598 . . . . 5 ((𝑧R𝑤R) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
5445, 48, 533eqtrrd 2841 . . . 4 ((𝑧R𝑤R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
55 opex 5324 . . . . 5 𝑧, 0R⟩ ∈ V
56 opex 5324 . . . . 5 𝑤, 0R⟩ ∈ V
57 eleq1 2880 . . . . . . 7 (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
58 eleq1 2880 . . . . . . 7 (𝑦 = ⟨𝑤, 0R⟩ → (𝑦 ∈ ℝ ↔ ⟨𝑤, 0R⟩ ∈ ℝ))
5957, 58bi2anan9 638 . . . . . 6 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ↔ (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ)))
60 oveq1 7146 . . . . . . . 8 (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
61 oveq2 7147 . . . . . . . . 9 (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
6261oveq2d 7155 . . . . . . . 8 (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
6360, 62sylan9eq 2856 . . . . . . 7 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
6463eqeq2d 2812 . . . . . 6 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
6559, 64anbi12d 633 . . . . 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 3559 . . . 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 587 . . 3 ((𝑧R𝑤R) → ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
68 r2ex 3265 . . 3 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
6967, 68sylibr 237 . 2 ((𝑧R𝑤R) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)))
701, 3, 69optocl 5613 1 (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  wrex 3110  cop 4534  (class class class)co 7139  Rcnr 10280  0Rc0r 10281  1Rc1r 10282  -1Rcm1r 10283   +R cplr 10284   ·R cmr 10285  cc 10528  cr 10529  ici 10532   + caddc 10533   · cmul 10535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-ec 8278  df-qs 8282  df-ni 10287  df-pli 10288  df-mi 10289  df-lti 10290  df-plpq 10323  df-mpq 10324  df-ltpq 10325  df-enq 10326  df-nq 10327  df-erq 10328  df-plq 10329  df-mq 10330  df-1nq 10331  df-rq 10332  df-ltnq 10333  df-np 10396  df-1p 10397  df-plp 10398  df-mp 10399  df-ltp 10400  df-enr 10470  df-nr 10471  df-plr 10472  df-mr 10473  df-0r 10475  df-1r 10476  df-m1r 10477  df-c 10536  df-i 10539  df-r 10540  df-add 10541  df-mul 10542
This theorem is referenced by: (None)
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