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Theorem cnref1o 9102
Description: There is a natural one-to-one mapping from (ℝ × ℝ) to , where we map 𝑥, 𝑦 to (𝑥 + (i · 𝑦)). In our construction of the complex numbers, this is in fact our definition of (see df-c 7335), but in the axiomatic treatment we can only show that there is the expected mapping between these two sets. (Contributed by Mario Carneiro, 16-Jun-2013.) (Revised by Mario Carneiro, 17-Feb-2014.)
Hypothesis
Ref Expression
cnref1o.1 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
Assertion
Ref Expression
cnref1o 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem cnref1o
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 107 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
21recnd 7495 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
3 ax-icn 7419 . . . . . . . . 9 i ∈ ℂ
43a1i 9 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
5 simpr 108 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
65recnd 7495 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
74, 6mulcld 7487 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
82, 7addcld 7486 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
98rgen2a 2429 . . . . 5 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
10 cnref1o.1 . . . . . 6 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
1110fnmpt2 5954 . . . . 5 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
129, 11ax-mp 7 . . . 4 𝐹 Fn (ℝ × ℝ)
13 1st2nd2 5927 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1413fveq2d 5293 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
15 df-ov 5637 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
1614, 15syl6eqr 2138 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
17 xp1st 5918 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
18 xp2nd 5919 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
1917recnd 7495 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℂ)
203a1i 9 . . . . . . . . . 10 (𝑧 ∈ (ℝ × ℝ) → i ∈ ℂ)
2118recnd 7495 . . . . . . . . . 10 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℂ)
2220, 21mulcld 7487 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → (i · (2nd𝑧)) ∈ ℂ)
2319, 22addcld 7486 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ)
24 oveq1 5641 . . . . . . . . 9 (𝑥 = (1st𝑧) → (𝑥 + (i · 𝑦)) = ((1st𝑧) + (i · 𝑦)))
25 oveq2 5642 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (i · 𝑦) = (i · (2nd𝑧)))
2625oveq2d 5650 . . . . . . . . 9 (𝑦 = (2nd𝑧) → ((1st𝑧) + (i · 𝑦)) = ((1st𝑧) + (i · (2nd𝑧))))
2724, 26, 10ovmpt2g 5761 . . . . . . . 8 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ ∧ ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
2817, 18, 23, 27syl3anc 1174 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
2916, 28eqtrd 2120 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))))
3029, 23eqeltrd 2164 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ ℂ)
3130rgen 2428 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ
32 ffnfv 5440 . . . 4 (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ))
3312, 31, 32mpbir2an 888 . . 3 𝐹:(ℝ × ℝ)⟶ℂ
3417, 18jca 300 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ))
35 xp1st 5918 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
36 xp2nd 5919 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
3735, 36jca 300 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ))
38 cru 8055 . . . . . . 7 ((((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) ∧ ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
3934, 37, 38syl2an 283 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
40 fveq2 5289 . . . . . . . . 9 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
41 fveq2 5289 . . . . . . . . . 10 (𝑧 = 𝑤 → (1st𝑧) = (1st𝑤))
42 fveq2 5289 . . . . . . . . . . 11 (𝑧 = 𝑤 → (2nd𝑧) = (2nd𝑤))
4342oveq2d 5650 . . . . . . . . . 10 (𝑧 = 𝑤 → (i · (2nd𝑧)) = (i · (2nd𝑤)))
4441, 43oveq12d 5652 . . . . . . . . 9 (𝑧 = 𝑤 → ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))))
4540, 44eqeq12d 2102 . . . . . . . 8 (𝑧 = 𝑤 → ((𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))) ↔ (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤)))))
4645, 29vtoclga 2685 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤))))
4729, 46eqeqan12d 2103 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤)))))
48 1st2nd2 5927 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4913, 48eqeqan12d 2103 . . . . . . 7 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
50 vex 2622 . . . . . . . . 9 𝑧 ∈ V
51 1stexg 5920 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
5250, 51ax-mp 7 . . . . . . . 8 (1st𝑧) ∈ V
53 2ndexg 5921 . . . . . . . . 9 (𝑧 ∈ V → (2nd𝑧) ∈ V)
5450, 53ax-mp 7 . . . . . . . 8 (2nd𝑧) ∈ V
5552, 54opth 4055 . . . . . . 7 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
5649, 55syl6bb 194 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
5739, 47, 563bitr4d 218 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5857biimpd 142 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
5958rgen2a 2429 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
60 dff13 5529 . . 3 (𝐹:(ℝ × ℝ)–1-1→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6133, 59, 60mpbir2an 888 . 2 𝐹:(ℝ × ℝ)–1-1→ℂ
62 cnre 7463 . . . . . 6 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
63 simpl 107 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑢 ∈ ℝ)
64 simpr 108 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ∈ ℝ)
6563recnd 7495 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑢 ∈ ℂ)
663a1i 9 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → i ∈ ℂ)
6764recnd 7495 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ∈ ℂ)
6866, 67mulcld 7487 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (i · 𝑣) ∈ ℂ)
6965, 68addcld 7486 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + (i · 𝑣)) ∈ ℂ)
70 oveq1 5641 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
71 oveq2 5642 . . . . . . . . . . 11 (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
7271oveq2d 5650 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
7370, 72, 10ovmpt2g 5761 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ (𝑢 + (i · 𝑣)) ∈ ℂ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
7463, 64, 69, 73syl3anc 1174 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
7574eqeq2d 2099 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
76752rexbiia 2394 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
7762, 76sylibr 132 . . . . 5 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
78 fveq2 5289 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
79 df-ov 5637 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
8078, 79syl6eqr 2138 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
8180eqeq2d 2099 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
8281rexxp 4568 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
8377, 82sylibr 132 . . . 4 (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
8483rgen 2428 . . 3 𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
85 dffo3 5430 . . 3 (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
8633, 84, 85mpbir2an 888 . 2 𝐹:(ℝ × ℝ)–onto→ℂ
87 df-f1o 5009 . 2 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
8861, 86, 87mpbir2an 888 1 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  wrex 2360  Vcvv 2619  cop 3444   × cxp 4426   Fn wfn 4997  wf 4998  1-1wf1 4999  ontowfo 5000  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  cmpt2 5636  1st c1st 5891  2nd c2nd 5892  cc 7327  cr 7328  ici 7331   + caddc 7332   · cmul 7334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-ltxr 7506  df-sub 7634  df-neg 7635  df-reap 8028
This theorem is referenced by:  cnrecnv  10309
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