ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnref1o GIF version

Theorem cnref1o 9716
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 7878), 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 109 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
21recnd 8048 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
3 ax-icn 7967 . . . . . . . . 9 i ∈ ℂ
43a1i 9 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
5 simpr 110 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
65recnd 8048 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
74, 6mulcld 8040 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
82, 7addcld 8039 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
98rgen2a 2548 . . . . 5 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
10 cnref1o.1 . . . . . 6 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
1110fnmpo 6255 . . . . 5 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
129, 11ax-mp 5 . . . 4 𝐹 Fn (ℝ × ℝ)
13 1st2nd2 6228 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1413fveq2d 5558 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
15 df-ov 5921 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
1614, 15eqtr4di 2244 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
17 xp1st 6218 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
18 xp2nd 6219 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
1917recnd 8048 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℂ)
203a1i 9 . . . . . . . . . 10 (𝑧 ∈ (ℝ × ℝ) → i ∈ ℂ)
2118recnd 8048 . . . . . . . . . 10 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℂ)
2220, 21mulcld 8040 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → (i · (2nd𝑧)) ∈ ℂ)
2319, 22addcld 8039 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ)
24 oveq1 5925 . . . . . . . . 9 (𝑥 = (1st𝑧) → (𝑥 + (i · 𝑦)) = ((1st𝑧) + (i · 𝑦)))
25 oveq2 5926 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (i · 𝑦) = (i · (2nd𝑧)))
2625oveq2d 5934 . . . . . . . . 9 (𝑦 = (2nd𝑧) → ((1st𝑧) + (i · 𝑦)) = ((1st𝑧) + (i · (2nd𝑧))))
2724, 26, 10ovmpog 6053 . . . . . . . 8 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ ∧ ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
2817, 18, 23, 27syl3anc 1249 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
2916, 28eqtrd 2226 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))))
3029, 23eqeltrd 2270 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ ℂ)
3130rgen 2547 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ
32 ffnfv 5716 . . . 4 (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ))
3312, 31, 32mpbir2an 944 . . 3 𝐹:(ℝ × ℝ)⟶ℂ
3417, 18jca 306 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ))
35 xp1st 6218 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
36 xp2nd 6219 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
3735, 36jca 306 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ))
38 cru 8621 . . . . . . 7 ((((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) ∧ ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
3934, 37, 38syl2an 289 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
40 fveq2 5554 . . . . . . . . 9 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
41 fveq2 5554 . . . . . . . . . 10 (𝑧 = 𝑤 → (1st𝑧) = (1st𝑤))
42 fveq2 5554 . . . . . . . . . . 11 (𝑧 = 𝑤 → (2nd𝑧) = (2nd𝑤))
4342oveq2d 5934 . . . . . . . . . 10 (𝑧 = 𝑤 → (i · (2nd𝑧)) = (i · (2nd𝑤)))
4441, 43oveq12d 5936 . . . . . . . . 9 (𝑧 = 𝑤 → ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))))
4540, 44eqeq12d 2208 . . . . . . . 8 (𝑧 = 𝑤 → ((𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))) ↔ (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤)))))
4645, 29vtoclga 2826 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤))))
4729, 46eqeqan12d 2209 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤)))))
48 1st2nd2 6228 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4913, 48eqeqan12d 2209 . . . . . . 7 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
50 vex 2763 . . . . . . . . 9 𝑧 ∈ V
51 1stexg 6220 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
5250, 51ax-mp 5 . . . . . . . 8 (1st𝑧) ∈ V
53 2ndexg 6221 . . . . . . . . 9 (𝑧 ∈ V → (2nd𝑧) ∈ V)
5450, 53ax-mp 5 . . . . . . . 8 (2nd𝑧) ∈ V
5552, 54opth 4266 . . . . . . 7 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
5649, 55bitrdi 196 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
5739, 47, 563bitr4d 220 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5857biimpd 144 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
5958rgen2a 2548 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
60 dff13 5811 . . 3 (𝐹:(ℝ × ℝ)–1-1→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6133, 59, 60mpbir2an 944 . 2 𝐹:(ℝ × ℝ)–1-1→ℂ
62 cnre 8015 . . . . . 6 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
63 simpl 109 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑢 ∈ ℝ)
64 simpr 110 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ∈ ℝ)
6563recnd 8048 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑢 ∈ ℂ)
663a1i 9 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → i ∈ ℂ)
6764recnd 8048 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ∈ ℂ)
6866, 67mulcld 8040 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (i · 𝑣) ∈ ℂ)
6965, 68addcld 8039 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + (i · 𝑣)) ∈ ℂ)
70 oveq1 5925 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
71 oveq2 5926 . . . . . . . . . . 11 (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
7271oveq2d 5934 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
7370, 72, 10ovmpog 6053 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ (𝑢 + (i · 𝑣)) ∈ ℂ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
7463, 64, 69, 73syl3anc 1249 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
7574eqeq2d 2205 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
76752rexbiia 2510 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
7762, 76sylibr 134 . . . . 5 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
78 fveq2 5554 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
79 df-ov 5921 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
8078, 79eqtr4di 2244 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
8180eqeq2d 2205 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
8281rexxp 4806 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
8377, 82sylibr 134 . . . 4 (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
8483rgen 2547 . . 3 𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
85 dffo3 5705 . . 3 (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
8633, 84, 85mpbir2an 944 . 2 𝐹:(ℝ × ℝ)–onto→ℂ
87 df-f1o 5261 . 2 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
8861, 86, 87mpbir2an 944 1 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  wral 2472  wrex 2473  Vcvv 2760  cop 3621   × cxp 4657   Fn wfn 5249  wf 5250  1-1wf1 5251  ontowfo 5252  1-1-ontowf1o 5253  cfv 5254  (class class class)co 5918  cmpo 5920  1st c1st 6191  2nd c2nd 6192  cc 7870  cr 7871  ici 7874   + caddc 7875   · cmul 7877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-sub 8192  df-neg 8193  df-reap 8594
This theorem is referenced by:  cnrecnv  11054
  Copyright terms: Public domain W3C validator