Theorem cnref1o 12725

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 10877), 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.

Step	Hyp	Ref	Expression
1		cnref1o.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
2		ovex 7308	. . . . 5 ⊢ (𝑥 + (i · 𝑦)) ∈ V
3	1, 2	fnmpoi 7910	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 7870	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5	4	fveq2d 6778	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
6		df-ov 7278	. . . . . . . 8 ⊢ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2796	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
8		xp1st 7863	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
9		xp2nd 7864	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
10		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + (i · 𝑦)) = ((1st ‘𝑧) + (i · 𝑦)))
11		oveq2 7283	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (i · 𝑦) = (i · (2nd ‘𝑧)))
12	11	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧) + (i · 𝑦)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
13		ovex 7308	. . . . . . . . 9 ⊢ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ V
14	10, 12, 1, 13	ovmpo 7433	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
15	8, 9, 14	syl2anc 584	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
16	7, 15	eqtrd 2778	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
17	8	recnd 11003	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℂ)
18		ax-icn 10930	. . . . . . . 8 ⊢ i ∈ ℂ
19	9	recnd 11003	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℂ)
20		mulcl 10955	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ (2nd ‘𝑧) ∈ ℂ) → (i · (2nd ‘𝑧)) ∈ ℂ)
21	18, 19, 20	sylancr 587	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (i · (2nd ‘𝑧)) ∈ ℂ)
22	17, 21	addcld 10994	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ ℂ)
23	16, 22	eqeltrd 2839	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ ℂ)
24	23	rgen 3074	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ
25		ffnfv 6992	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ))
26	3, 24, 25	mpbir2an 708	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶ℂ
27	8, 9	jca 512	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ))
28		xp1st 7863	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1st ‘𝑤) ∈ ℝ)
29		xp2nd 7864	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2nd ‘𝑤) ∈ ℝ)
30	28, 29	jca 512	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1st ‘𝑤) ∈ ℝ ∧ (2nd ‘𝑤) ∈ ℝ))
31		cru 11965	. . . . . . 7 ⊢ ((((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) ∧ ((1st ‘𝑤) ∈ ℝ ∧ (2nd ‘𝑤) ∈ ℝ)) → (((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))) ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤))))
32	27, 30, 31	syl2an 596	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))) ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤))))
33		fveq2 6774	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
34		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
35		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (2nd ‘𝑧) = (2nd ‘𝑤))
36	35	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (i · (2nd ‘𝑧)) = (i · (2nd ‘𝑤)))
37	34, 36	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
38	33, 37	eqeq12d 2754	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ↔ (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
39	38, 16	vtoclga 3513	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
40	16, 39	eqeqan12d 2752	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
41		1st2nd2 7870	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
42	4, 41	eqeqan12d 2752	. . . . . . 7 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
43		fvex 6787	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
44		fvex 6787	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45	43, 44	opth 5391	. . . . . . 7 ⊢ (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤)))
46	42, 45	bitrdi 287	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤))))
47	32, 40, 46	3bitr4d 311	. . . . 5 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
48	47	biimpd 228	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
49	48	rgen2 3120	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
50		dff13 7128	. . 3 ⊢ (𝐹:(ℝ × ℝ)–1-1→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
51	26, 49, 50	mpbir2an 708	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→ℂ
52		cnre 10972	. . . . . 6 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
53		oveq1 7282	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
54		oveq2 7283	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
55	54	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
56		ovex 7308	. . . . . . . . 9 ⊢ (𝑢 + (i · 𝑣)) ∈ V
57	53, 55, 1, 56	ovmpo 7433	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
58	57	eqeq2d 2749	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
59	58	2rexbiia 3227	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
60	52, 59	sylibr 233	. . . . 5 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
61		fveq2 6774	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
62		df-ov 7278	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
63	61, 62	eqtr4di 2796	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
64	63	eqeq2d 2749	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
65	64	rexxp 5751	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
66	60, 65	sylibr 233	. . . 4 ⊢ (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
67	66	rgen 3074	. . 3 ⊢ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
68		dffo3 6978	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
69	26, 67, 68	mpbir2an 708	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→ℂ
70		df-f1o 6440	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
71	51, 69, 70	mpbir2an 708	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  ⟨cop 4567   × cxp 5587   Fn wfn 6428  ⟶wf 6429  –1-1→wf1 6430  –onto→wfo 6431  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830  ℂcc 10869  ℝcr 10870  ici 10873   + caddc 10874   · cmul 10876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633

This theorem is referenced by:  cnexALT  12726  cnrecnv  14876  cpnnen  15938  cnheiborlem  24117  mbfimaopnlem  24819