Theorem cnref1o 12920

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 11050), 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 7402	. . . . 5 ⊢ (𝑥 + (i · 𝑦)) ∈ V
3	1, 2	fnmpoi 8028	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 7986	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5	4	fveq2d 6844	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
6		df-ov 7372	. . . . . . . 8 ⊢ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2782	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
8		xp1st 7979	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
9		xp2nd 7980	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
10		oveq1 7376	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + (i · 𝑦)) = ((1st ‘𝑧) + (i · 𝑦)))
11		oveq2 7377	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (i · 𝑦) = (i · (2nd ‘𝑧)))
12	11	oveq2d 7385	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧) + (i · 𝑦)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
13		ovex 7402	. . . . . . . . 9 ⊢ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ V
14	10, 12, 1, 13	ovmpo 7529	. . . . . . . 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 2764	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
17	8	recnd 11178	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℂ)
18		ax-icn 11103	. . . . . . . 8 ⊢ i ∈ ℂ
19	9	recnd 11178	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℂ)
20		mulcl 11128	. . . . . . . 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 11169	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ ℂ)
23	16, 22	eqeltrd 2828	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ ℂ)
24	23	rgen 3046	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ
25		ffnfv 7073	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ))
26	3, 24, 25	mpbir2an 711	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶ℂ
27	8, 9	jca 511	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ))
28		xp1st 7979	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1st ‘𝑤) ∈ ℝ)
29		xp2nd 7980	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2nd ‘𝑤) ∈ ℝ)
30	28, 29	jca 511	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1st ‘𝑤) ∈ ℝ ∧ (2nd ‘𝑤) ∈ ℝ))
31		cru 12154	. . . . . . 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 6840	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
34		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
35		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (2nd ‘𝑧) = (2nd ‘𝑤))
36	35	oveq2d 7385	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (i · (2nd ‘𝑧)) = (i · (2nd ‘𝑤)))
37	34, 36	oveq12d 7387	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
38	33, 37	eqeq12d 2745	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ↔ (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
39	38, 16	vtoclga 3540	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
40	16, 39	eqeqan12d 2743	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
41		1st2nd2 7986	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
42	4, 41	eqeqan12d 2743	. . . . . . 7 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
43		fvex 6853	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
44		fvex 6853	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45	43, 44	opth 5431	. . . . . . 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 229	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
49	48	rgen2 3175	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
50		dff13 7211	. . 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 711	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→ℂ
52		cnre 11147	. . . . . 6 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
53		oveq1 7376	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
54		oveq2 7377	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
55	54	oveq2d 7385	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
56		ovex 7402	. . . . . . . . 9 ⊢ (𝑢 + (i · 𝑣)) ∈ V
57	53, 55, 1, 56	ovmpo 7529	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
58	57	eqeq2d 2740	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
59	58	2rexbiia 3196	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
60	52, 59	sylibr 234	. . . . 5 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
61		fveq2 6840	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
62		df-ov 7372	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
63	61, 62	eqtr4di 2782	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
64	63	eqeq2d 2740	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
65	64	rexxp 5796	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
66	60, 65	sylibr 234	. . . 4 ⊢ (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
67	66	rgen 3046	. . 3 ⊢ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
68		dffo3 7056	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
69	26, 67, 68	mpbir2an 711	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→ℂ
70		df-f1o 6506	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
71	51, 69, 70	mpbir2an 711	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ⟨cop 4591   × cxp 5629   Fn wfn 6494  ⟶wf 6495  –1-1→wf1 6496  –onto→wfo 6497  –1-1-onto→wf1o 6498  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371  1^st c1st 7945  2^nd c2nd 7946  ℂcc 11042  ℝcr 11043  ici 11046   + caddc 11047   · cmul 11049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812

This theorem is referenced by:  cnexALT  12921  cnrecnv  15107  cpnnen  16173  cnheiborlem  24886  mbfimaopnlem  25589