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Theorem cnref1o 9496
 Description: There is a natural one-to-one mapping from (ℝ × ℝ) to ℂ, where we map ⟨𝑥, 𝑦⟩ to (𝑥 + (i · 𝑦)). In our construction of the complex numbers, this is in fact our definition of ℂ (see df-c 7677), 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 108 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
21recnd 7845 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
3 ax-icn 7766 . . . . . . . . 9 i ∈ ℂ
43a1i 9 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
5 simpr 109 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
65recnd 7845 . . . . . . . 8 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
74, 6mulcld 7837 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (i · 𝑦) ∈ ℂ)
82, 7addcld 7836 . . . . . 6 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + (i · 𝑦)) ∈ ℂ)
98rgen2a 2490 . . . . 5 𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ
10 cnref1o.1 . . . . . 6 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
1110fnmpo 6111 . . . . 5 (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥 + (i · 𝑦)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
129, 11ax-mp 5 . . . 4 𝐹 Fn (ℝ × ℝ)
13 1st2nd2 6084 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1413fveq2d 5436 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
15 df-ov 5788 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
1614, 15eqtr4di 2191 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
17 xp1st 6074 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
18 xp2nd 6075 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
1917recnd 7845 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℂ)
203a1i 9 . . . . . . . . . 10 (𝑧 ∈ (ℝ × ℝ) → i ∈ ℂ)
2118recnd 7845 . . . . . . . . . 10 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℂ)
2220, 21mulcld 7837 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → (i · (2nd𝑧)) ∈ ℂ)
2319, 22addcld 7836 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ)
24 oveq1 5792 . . . . . . . . 9 (𝑥 = (1st𝑧) → (𝑥 + (i · 𝑦)) = ((1st𝑧) + (i · 𝑦)))
25 oveq2 5793 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → (i · 𝑦) = (i · (2nd𝑧)))
2625oveq2d 5801 . . . . . . . . 9 (𝑦 = (2nd𝑧) → ((1st𝑧) + (i · 𝑦)) = ((1st𝑧) + (i · (2nd𝑧))))
2724, 26, 10ovmpog 5916 . . . . . . . 8 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ ∧ ((1st𝑧) + (i · (2nd𝑧))) ∈ ℂ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
2817, 18, 23, 27syl3anc 1217 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = ((1st𝑧) + (i · (2nd𝑧))))
2916, 28eqtrd 2173 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))))
3029, 23eqeltrd 2217 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ ℂ)
3130rgen 2489 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ
32 ffnfv 5589 . . . 4 (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ ℂ))
3312, 31, 32mpbir2an 927 . . 3 𝐹:(ℝ × ℝ)⟶ℂ
3417, 18jca 304 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ))
35 xp1st 6074 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
36 xp2nd 6075 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
3735, 36jca 304 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ))
38 cru 8415 . . . . . . 7 ((((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) ∧ ((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
3934, 37, 38syl2an 287 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))) ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
40 fveq2 5432 . . . . . . . . 9 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
41 fveq2 5432 . . . . . . . . . 10 (𝑧 = 𝑤 → (1st𝑧) = (1st𝑤))
42 fveq2 5432 . . . . . . . . . . 11 (𝑧 = 𝑤 → (2nd𝑧) = (2nd𝑤))
4342oveq2d 5801 . . . . . . . . . 10 (𝑧 = 𝑤 → (i · (2nd𝑧)) = (i · (2nd𝑤)))
4441, 43oveq12d 5803 . . . . . . . . 9 (𝑧 = 𝑤 → ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤))))
4540, 44eqeq12d 2155 . . . . . . . 8 (𝑧 = 𝑤 → ((𝐹𝑧) = ((1st𝑧) + (i · (2nd𝑧))) ↔ (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤)))))
4645, 29vtoclga 2756 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤) + (i · (2nd𝑤))))
4729, 46eqeqan12d 2156 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ((1st𝑧) + (i · (2nd𝑧))) = ((1st𝑤) + (i · (2nd𝑤)))))
48 1st2nd2 6084 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4913, 48eqeqan12d 2156 . . . . . . 7 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
50 vex 2693 . . . . . . . . 9 𝑧 ∈ V
51 1stexg 6076 . . . . . . . . 9 (𝑧 ∈ V → (1st𝑧) ∈ V)
5250, 51ax-mp 5 . . . . . . . 8 (1st𝑧) ∈ V
53 2ndexg 6077 . . . . . . . . 9 (𝑧 ∈ V → (2nd𝑧) ∈ V)
5450, 53ax-mp 5 . . . . . . . 8 (2nd𝑧) ∈ V
5552, 54opth 4169 . . . . . . 7 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
5649, 55syl6bb 195 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
5739, 47, 563bitr4d 219 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ 𝑧 = 𝑤))
5857biimpd 143 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
5958rgen2a 2490 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
60 dff13 5680 . . 3 (𝐹:(ℝ × ℝ)–1-1→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
6133, 59, 60mpbir2an 927 . 2 𝐹:(ℝ × ℝ)–1-1→ℂ
62 cnre 7813 . . . . . 6 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
63 simpl 108 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑢 ∈ ℝ)
64 simpr 109 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ∈ ℝ)
6563recnd 7845 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑢 ∈ ℂ)
663a1i 9 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → i ∈ ℂ)
6764recnd 7845 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝑣 ∈ ℂ)
6866, 67mulcld 7837 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (i · 𝑣) ∈ ℂ)
6965, 68addcld 7836 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + (i · 𝑣)) ∈ ℂ)
70 oveq1 5792 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
71 oveq2 5793 . . . . . . . . . . 11 (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
7271oveq2d 5801 . . . . . . . . . 10 (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
7370, 72, 10ovmpog 5916 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ (𝑢 + (i · 𝑣)) ∈ ℂ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
7463, 64, 69, 73syl3anc 1217 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
7574eqeq2d 2152 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
76752rexbiia 2455 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
7762, 76sylibr 133 . . . . 5 (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
78 fveq2 5432 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
79 df-ov 5788 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
8078, 79eqtr4di 2191 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
8180eqeq2d 2152 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
8281rexxp 4694 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
8377, 82sylibr 133 . . . 4 (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
8483rgen 2489 . . 3 𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
85 dffo3 5578 . . 3 (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
8633, 84, 85mpbir2an 927 . 2 𝐹:(ℝ × ℝ)–onto→ℂ
87 df-f1o 5141 . 2 (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
8861, 86, 87mpbir2an 927 1 𝐹:(ℝ × ℝ)–1-1-onto→ℂ
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2690  ⟨cop 3536   × cxp 4548   Fn wfn 5129  ⟶wf 5130  –1-1→wf1 5131  –onto→wfo 5132  –1-1-onto→wf1o 5133  ‘cfv 5134  (class class class)co 5785   ∈ cmpo 5787  1st c1st 6047  2nd c2nd 6048  ℂcc 7669  ℝcr 7670  ici 7673   + caddc 7674   · cmul 7676 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462  ax-cnex 7762  ax-resscn 7763  ax-1cn 7764  ax-1re 7765  ax-icn 7766  ax-addcl 7767  ax-addrcl 7768  ax-mulcl 7769  ax-mulrcl 7770  ax-addcom 7771  ax-mulcom 7772  ax-addass 7773  ax-mulass 7774  ax-distr 7775  ax-i2m1 7776  ax-0lt1 7777  ax-1rid 7778  ax-0id 7779  ax-rnegex 7780  ax-precex 7781  ax-cnre 7782  ax-pre-ltirr 7783  ax-pre-lttrn 7785  ax-pre-apti 7786  ax-pre-ltadd 7787  ax-pre-mulgt0 7788 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-f1 5139  df-fo 5140  df-f1o 5141  df-fv 5142  df-riota 5741  df-ov 5788  df-oprab 5789  df-mpo 5790  df-1st 6049  df-2nd 6050  df-pnf 7853  df-mnf 7854  df-ltxr 7856  df-sub 7986  df-neg 7987  df-reap 8388 This theorem is referenced by:  cnrecnv  10741
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