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Theorem cnref1o 11659
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 9798), 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 cnref1o.1 . . . . 5 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
2 ovex 6555 . . . . 5 (𝑥 + (i · 𝑦)) ∈ V
31, 2fnmpt2i 7105 . . . 4 𝐹 Fn (ℝ × ℝ)
4 1st2nd2 7073 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
54fveq2d 6092 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
6 df-ov 6530 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
75, 6syl6eqr 2661 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
8 xp1st 7066 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
9 xp2nd 7067 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
10 oveq1 6534 . . . . . . . . 9 (𝑥 = (1st𝑧) → (𝑥 + (i · 𝑦)) = ((1st𝑧) + (i · 𝑦)))
11 oveq2 6535 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (i · 𝑦) = (i · (2nd𝑧)))
1211oveq2d 6543 . . . . . . . . 9 (𝑦 = (2nd𝑧) → ((1st𝑧) + (i · 𝑦)) = ((1st𝑧) + (i · (2nd𝑧))))
13 ovex 6555 . . . . . . . . 9 ((1st𝑧) + (i · (2nd𝑧))) ∈ V
1410, 12, 1, 13ovmpt2 6672 . . . . . . . 8 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
158, 9, 14syl2anc 690 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
167, 15eqtrd 2643 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))))
178recnd 9924 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℂ)
18 ax-icn 9851 . . . . . . . 8 i ∈ ℂ
199recnd 9924 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℂ)
20 mulcl 9876 . . . . . . . 8 ((i ∈ ℂ ∧ (2nd𝑧) ∈ ℂ) → (i · (2nd𝑧)) ∈ ℂ)
2118, 19, 20sylancr 693 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (i · (2nd𝑧)) ∈ ℂ)
2217, 21addcld 9915 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ)
2316, 22eqeltrd 2687 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ ℂ)
2423rgen 2905 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ
25 ffnfv 6280 . . . 4 (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ))
263, 24, 25mpbir2an 956 . . 3 𝐹:(ℝ × ℝ)⟶ℂ
278, 9jca 552 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ))
28 xp1st 7066 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
29 xp2nd 7067 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
3028, 29jca 552 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ))
31 cru 10859 . . . . . . 7 ((((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) ∧ ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
3227, 30, 31syl2an 492 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
33 fveq2 6088 . . . . . . . . 9 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
34 fveq2 6088 . . . . . . . . . 10 (𝑧 = 𝑤 → (1st𝑧) = (1st𝑤))
35 fveq2 6088 . . . . . . . . . . 11 (𝑧 = 𝑤 → (2nd𝑧) = (2nd𝑤))
3635oveq2d 6543 . . . . . . . . . 10 (𝑧 = 𝑤 → (i · (2nd𝑧)) = (i · (2nd𝑤)))
3734, 36oveq12d 6545 . . . . . . . . 9 (𝑧 = 𝑤 → ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))))
3833, 37eqeq12d 2624 . . . . . . . 8 (𝑧 = 𝑤 → ((𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))) ↔ (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤)))))
3938, 16vtoclga 3244 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤))))
4016, 39eqeqan12d 2625 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤)))))
41 1st2nd2 7073 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
424, 41eqeqan12d 2625 . . . . . . 7 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
43 fvex 6098 . . . . . . . 8 (1st𝑧) ∈ V
44 fvex 6098 . . . . . . . 8 (2nd𝑧) ∈ V
4543, 44opth 4865 . . . . . . 7 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
4642, 45syl6bb 274 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4732, 40, 463bitr4d 298 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
4847biimpd 217 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4948rgen2a 2959 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
50 dff13 6394 . . 3 (𝐹:(ℝ × ℝ)–1-1→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
5126, 49, 50mpbir2an 956 . 2 𝐹:(ℝ × ℝ)–1-1→ℂ
52 cnre 9892 . . . . . 6 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
53 oveq1 6534 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
54 oveq2 6535 . . . . . . . . . 10 (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
5554oveq2d 6543 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
56 ovex 6555 . . . . . . . . 9 (𝑢 + (i · 𝑣)) ∈ V
5753, 55, 1, 56ovmpt2 6672 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
5857eqeq2d 2619 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
59582rexbiia 3036 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
6052, 59sylibr 222 . . . . 5 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
61 fveq2 6088 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
62 df-ov 6530 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
6361, 62syl6eqr 2661 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
6463eqeq2d 2619 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
6564rexxp 5174 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
6660, 65sylibr 222 . . . 4 (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
6766rgen 2905 . . 3 𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
68 dffo3 6267 . . 3 (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
6926, 67, 68mpbir2an 956 . 2 𝐹:(ℝ × ℝ)–onto→ℂ
70 df-f1o 5797 . 2 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
7151, 69, 70mpbir2an 956 1 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  wrex 2896  cop 4130   × cxp 5026   Fn wfn 5785  wf 5786  1-1wf1 5787  ontowfo 5788  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  cc 9790  cr 9791  ici 9794   + caddc 9795   · cmul 9797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-po 4949  df-so 4950  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534
This theorem is referenced by:  cnexALT  11660  cnrecnv  13699  cpnnen  14743  cnheiborlem  22492  mbfimaopnlem  23145
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