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Theorem axcnre 11204
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 11228. (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 11161 . 2 ℂ = (R × R)
2 eqeq1 2741 . . 3 (⟨𝑧, 𝑤⟩ = 𝐴 → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ 𝐴 = (𝑥 + (i · 𝑦))))
322rexbidv 3222 . 2 (⟨𝑧, 𝑤⟩ = 𝐴 → (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))))
4 opelreal 11170 . . . . . 6 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
5 opelreal 11170 . . . . . 6 (⟨𝑤, 0R⟩ ∈ ℝ ↔ 𝑤R)
64, 5anbi12i 628 . . . . 5 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ) ↔ (𝑧R𝑤R))
76biimpri 228 . . . 4 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ ∈ ℝ ∧ ⟨𝑤, 0R⟩ ∈ ℝ))
8 df-i 11164 . . . . . . . . 9 i = ⟨0R, 1R
98oveq1i 7441 . . . . . . . 8 (i · ⟨𝑤, 0R⟩) = (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩)
10 0r 11120 . . . . . . . . . 10 0RR
11 1sr 11121 . . . . . . . . . . 11 1RR
12 mulcnsr 11176 . . . . . . . . . . 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 702 . . . . . . . . . 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 691 . . . . . . . . 9 (𝑤R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩)
15 mulcomsr 11129 . . . . . . . . . . . . 13 (0R ·R 𝑤) = (𝑤 ·R 0R)
16 00sr 11139 . . . . . . . . . . . . 13 (𝑤R → (𝑤 ·R 0R) = 0R)
1715, 16eqtrid 2789 . . . . . . . . . . . 12 (𝑤R → (0R ·R 𝑤) = 0R)
1817oveq1d 7446 . . . . . . . . . . 11 (𝑤R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = (0R +R (-1R ·R (1R ·R 0R))))
19 00sr 11139 . . . . . . . . . . . . . . . 16 (1RR → (1R ·R 0R) = 0R)
2011, 19ax-mp 5 . . . . . . . . . . . . . . 15 (1R ·R 0R) = 0R
2120oveq2i 7442 . . . . . . . . . . . . . 14 (-1R ·R (1R ·R 0R)) = (-1R ·R 0R)
22 m1r 11122 . . . . . . . . . . . . . . 15 -1RR
23 00sr 11139 . . . . . . . . . . . . . . 15 (-1RR → (-1R ·R 0R) = 0R)
2422, 23ax-mp 5 . . . . . . . . . . . . . 14 (-1R ·R 0R) = 0R
2521, 24eqtri 2765 . . . . . . . . . . . . 13 (-1R ·R (1R ·R 0R)) = 0R
2625oveq2i 7442 . . . . . . . . . . . 12 (0R +R (-1R ·R (1R ·R 0R))) = (0R +R 0R)
27 0idsr 11137 . . . . . . . . . . . . 13 (0RR → (0R +R 0R) = 0R)
2810, 27ax-mp 5 . . . . . . . . . . . 12 (0R +R 0R) = 0R
2926, 28eqtri 2765 . . . . . . . . . . 11 (0R +R (-1R ·R (1R ·R 0R))) = 0R
3018, 29eqtrdi 2793 . . . . . . . . . 10 (𝑤R → ((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))) = 0R)
31 mulcomsr 11129 . . . . . . . . . . . . 13 (1R ·R 𝑤) = (𝑤 ·R 1R)
32 1idsr 11138 . . . . . . . . . . . . 13 (𝑤R → (𝑤 ·R 1R) = 𝑤)
3331, 32eqtrid 2789 . . . . . . . . . . . 12 (𝑤R → (1R ·R 𝑤) = 𝑤)
3433oveq1d 7446 . . . . . . . . . . 11 (𝑤R → ((1R ·R 𝑤) +R (0R ·R 0R)) = (𝑤 +R (0R ·R 0R)))
35 00sr 11139 . . . . . . . . . . . . . 14 (0RR → (0R ·R 0R) = 0R)
3610, 35ax-mp 5 . . . . . . . . . . . . 13 (0R ·R 0R) = 0R
3736oveq2i 7442 . . . . . . . . . . . 12 (𝑤 +R (0R ·R 0R)) = (𝑤 +R 0R)
38 0idsr 11137 . . . . . . . . . . . 12 (𝑤R → (𝑤 +R 0R) = 𝑤)
3937, 38eqtrid 2789 . . . . . . . . . . 11 (𝑤R → (𝑤 +R (0R ·R 0R)) = 𝑤)
4034, 39eqtrd 2777 . . . . . . . . . 10 (𝑤R → ((1R ·R 𝑤) +R (0R ·R 0R)) = 𝑤)
4130, 40opeq12d 4881 . . . . . . . . 9 (𝑤R → ⟨((0R ·R 𝑤) +R (-1R ·R (1R ·R 0R))), ((1R ·R 𝑤) +R (0R ·R 0R))⟩ = ⟨0R, 𝑤⟩)
4214, 41eqtrd 2777 . . . . . . . 8 (𝑤R → (⟨0R, 1R⟩ · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
439, 42eqtrid 2789 . . . . . . 7 (𝑤R → (i · ⟨𝑤, 0R⟩) = ⟨0R, 𝑤⟩)
4443oveq2d 7447 . . . . . 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 11175 . . . . . . 7 (((𝑧R ∧ 0RR) ∧ (0RR𝑤R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
4710, 46mpanl2 701 . . . . . 6 ((𝑧R ∧ (0RR𝑤R)) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
4810, 47mpanr1 703 . . . . 5 ((𝑧R𝑤R) → (⟨𝑧, 0R⟩ + ⟨0R, 𝑤⟩) = ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩)
49 0idsr 11137 . . . . . 6 (𝑧R → (𝑧 +R 0R) = 𝑧)
50 addcomsr 11127 . . . . . . 7 (0R +R 𝑤) = (𝑤 +R 0R)
5150, 38eqtrid 2789 . . . . . 6 (𝑤R → (0R +R 𝑤) = 𝑤)
52 opeq12 4875 . . . . . 6 (((𝑧 +R 0R) = 𝑧 ∧ (0R +R 𝑤) = 𝑤) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
5349, 51, 52syl2an 596 . . . . 5 ((𝑧R𝑤R) → ⟨(𝑧 +R 0R), (0R +R 𝑤)⟩ = ⟨𝑧, 𝑤⟩)
5445, 48, 533eqtrrd 2782 . . . 4 ((𝑧R𝑤R) → ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
55 opex 5469 . . . . 5 𝑧, 0R⟩ ∈ V
56 opex 5469 . . . . 5 𝑤, 0R⟩ ∈ V
57 eleq1 2829 . . . . . . 7 (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 ∈ ℝ ↔ ⟨𝑧, 0R⟩ ∈ ℝ))
58 eleq1 2829 . . . . . . 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 7438 . . . . . . . 8 (𝑥 = ⟨𝑧, 0R⟩ → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · 𝑦)))
61 oveq2 7439 . . . . . . . . 9 (𝑦 = ⟨𝑤, 0R⟩ → (i · 𝑦) = (i · ⟨𝑤, 0R⟩))
6261oveq2d 7447 . . . . . . . 8 (𝑦 = ⟨𝑤, 0R⟩ → (⟨𝑧, 0R⟩ + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
6360, 62sylan9eq 2797 . . . . . . 7 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (𝑥 + (i · 𝑦)) = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩)))
6463eqeq2d 2748 . . . . . 6 ((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) → (⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ⟨𝑧, 𝑤⟩ = (⟨𝑧, 0R⟩ + (i · ⟨𝑤, 0R⟩))))
6559, 64anbi12d 632 . . . . 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 3607 . . . 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 584 . . 3 ((𝑧R𝑤R) → ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
68 r2ex 3196 . . 3 (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)) ↔ ∃𝑥𝑦((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦))))
6967, 68sylibr 234 . 2 ((𝑧R𝑤R) → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ ⟨𝑧, 𝑤⟩ = (𝑥 + (i · 𝑦)))
701, 3, 69optocl 5780 1 (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cop 4632  (class class class)co 7431  Rcnr 10905  0Rc0r 10906  1Rc1r 10907  -1Rcm1r 10908   +R cplr 10909   ·R cmr 10910  cc 11153  cr 11154  ici 11157   + caddc 11158   · cmul 11160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-mq 10955  df-1nq 10956  df-rq 10957  df-ltnq 10958  df-np 11021  df-1p 11022  df-plp 11023  df-mp 11024  df-ltp 11025  df-enr 11095  df-nr 11096  df-plr 11097  df-mr 11098  df-0r 11100  df-1r 11101  df-m1r 11102  df-c 11161  df-i 11164  df-r 11165  df-add 11166  df-mul 11167
This theorem is referenced by: (None)
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