Theorem cnref1o 12986

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 11079), 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 7429	. . . . 5 ⊢ (𝑥 + (i · 𝑦)) ∈ V
3	1, 2	fnmpoi 8051	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 8009	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5	4	fveq2d 6871	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
6		df-ov 7399	. . . . . . . 8 ⊢ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2815	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
8		xp1st 8002	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
9		xp2nd 8003	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
10		oveq1 7403	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + (i · 𝑦)) = ((1st ‘𝑧) + (i · 𝑦)))
11		oveq2 7404	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (i · 𝑦) = (i · (2nd ‘𝑧)))
12	11	oveq2d 7412	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧) + (i · 𝑦)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
13		ovex 7429	. . . . . . . . 9 ⊢ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ V
14	10, 12, 1, 13	ovmpo 7556	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
15	8, 9, 14	syl2anc 593	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
16	7, 15	eqtrd 2797	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
17	8	recnd 11210	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℂ)
18		ax-icn 11132	. . . . . . . 8 ⊢ i ∈ ℂ
19	9	recnd 11210	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℂ)
20		mulcl 11157	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ (2nd ‘𝑧) ∈ ℂ) → (i · (2nd ‘𝑧)) ∈ ℂ)
21	18, 19, 20	sylancr 596	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (i · (2nd ‘𝑧)) ∈ ℂ)
22	17, 21	addcld 11201	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ ℂ)
23	16, 22	eqeltrd 2862	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ ℂ)
24	23	rgen 3078	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ
25		ffnfv 7100	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ))
26	3, 24, 25	mpbir2an 721	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶ℂ
27	8, 9	jca 519	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ))
28		xp1st 8002	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1st ‘𝑤) ∈ ℝ)
29		xp2nd 8003	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2nd ‘𝑤) ∈ ℝ)
30	28, 29	jca 519	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1st ‘𝑤) ∈ ℝ ∧ (2nd ‘𝑤) ∈ ℝ))
31		cru 12187	. . . . . . 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 605	. . . . . 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 6867	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
34		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
35		fveq2 6867	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (2nd ‘𝑧) = (2nd ‘𝑤))
36	35	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (i · (2nd ‘𝑧)) = (i · (2nd ‘𝑤)))
37	34, 36	oveq12d 7414	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
38	33, 37	eqeq12d 2778	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ↔ (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
39	38, 16	vtoclga 3541	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
40	16, 39	eqeqan12d 2776	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
41		1st2nd2 8009	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
42	4, 41	eqeqan12d 2776	. . . . . . 7 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
43		fvex 6880	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
44		fvex 6880	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45	43, 44	opth 5444	. . . . . . 7 ⊢ (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤)))
46	42, 45	bitrdi 289	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤))))
47	32, 40, 46	3bitr4d 313	. . . . 5 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
48	47	biimpd 231	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
49	48	rgen2 3202	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
50		dff13 7238	. . 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 721	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→ℂ
52		cnre 11178	. . . . . 6 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
53		oveq1 7403	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
54		oveq2 7404	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
55	54	oveq2d 7412	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
56		ovex 7429	. . . . . . . . 9 ⊢ (𝑢 + (i · 𝑣)) ∈ V
57	53, 55, 1, 56	ovmpo 7556	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
58	57	eqeq2d 2773	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
59	58	2rexbiia 3223	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
60	52, 59	sylibr 236	. . . . 5 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
61		fveq2 6867	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
62		df-ov 7399	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
63	61, 62	eqtr4di 2815	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
64	63	eqeq2d 2773	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
65	64	rexxp 5814	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
66	60, 65	sylibr 236	. . . 4 ⊢ (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
67	66	rgen 3078	. . 3 ⊢ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
68		dffo3 7083	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
69	26, 67, 68	mpbir2an 721	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→ℂ
70		df-f1o 6528	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
71	51, 69, 70	mpbir2an 721	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  ⟨cop 4588   × cxp 5645   Fn wfn 6516  ⟶wf 6517  –1-1→wf1 6518  –onto→wfo 6519  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396   ∈ cmpo 7398  1^st c1st 7968  2^nd c2nd 7969  ℂcc 11071  ℝcr 11072  ici 11075   + caddc 11076   · cmul 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845

This theorem is referenced by:  cnexALT  12987  cnrecnv  15192  cpnnen  16261  cnheiborlem  25016  mbfimaopnlem  25717