Theorem cnref1o 12438

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 10594), 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 7189	. . . . 5 ⊢ (𝑥 + (i · 𝑦)) ∈ V
3	1, 2	fnmpoi 7778	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 7738	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5	4	fveq2d 6667	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩))
6		df-ov 7159	. . . . . . . 8 ⊢ ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = (𝐹‘⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2811	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧)𝐹(2nd ‘𝑧)))
8		xp1st 7731	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
9		xp2nd 7732	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℝ)
10		oveq1 7163	. . . . . . . . 9 ⊢ (𝑥 = (1st ‘𝑧) → (𝑥 + (i · 𝑦)) = ((1st ‘𝑧) + (i · 𝑦)))
11		oveq2 7164	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑧) → (i · 𝑦) = (i · (2nd ‘𝑧)))
12	11	oveq2d 7172	. . . . . . . . 9 ⊢ (𝑦 = (2nd ‘𝑧) → ((1st ‘𝑧) + (i · 𝑦)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
13		ovex 7189	. . . . . . . . 9 ⊢ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ V
14	10, 12, 1, 13	ovmpo 7311	. . . . . . . 8 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
15	8, 9, 14	syl2anc 587	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧)𝐹(2nd ‘𝑧)) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
16	7, 15	eqtrd 2793	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))))
17	8	recnd 10720	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℂ)
18		ax-icn 10647	. . . . . . . 8 ⊢ i ∈ ℂ
19	9	recnd 10720	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2nd ‘𝑧) ∈ ℂ)
20		mulcl 10672	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ (2nd ‘𝑧) ∈ ℂ) → (i · (2nd ‘𝑧)) ∈ ℂ)
21	18, 19, 20	sylancr 590	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (i · (2nd ‘𝑧)) ∈ ℂ)
22	17, 21	addcld 10711	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ∈ ℂ)
23	16, 22	eqeltrd 2852	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ ℂ)
24	23	rgen 3080	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ
25		ffnfv 6879	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ ℂ))
26	3, 24, 25	mpbir2an 710	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶ℂ
27	8, 9	jca 515	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ))
28		xp1st 7731	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1st ‘𝑤) ∈ ℝ)
29		xp2nd 7732	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2nd ‘𝑤) ∈ ℝ)
30	28, 29	jca 515	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1st ‘𝑤) ∈ ℝ ∧ (2nd ‘𝑤) ∈ ℝ))
31		cru 11679	. . . . . . 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 598	. . . . . 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 6663	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
34		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (1st ‘𝑧) = (1st ‘𝑤))
35		fveq2 6663	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (2nd ‘𝑧) = (2nd ‘𝑤))
36	35	oveq2d 7172	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (i · (2nd ‘𝑧)) = (i · (2nd ‘𝑤)))
37	34, 36	oveq12d 7174	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
38	33, 37	eqeq12d 2774	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = ((1st ‘𝑧) + (i · (2nd ‘𝑧))) ↔ (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
39	38, 16	vtoclga 3494	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1st ‘𝑤) + (i · (2nd ‘𝑤))))
40	16, 39	eqeqan12d 2775	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ((1st ‘𝑧) + (i · (2nd ‘𝑧))) = ((1st ‘𝑤) + (i · (2nd ‘𝑤)))))
41		1st2nd2 7738	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
42	4, 41	eqeqan12d 2775	. . . . . . 7 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩))
43		fvex 6676	. . . . . . . 8 ⊢ (1st ‘𝑧) ∈ V
44		fvex 6676	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45	43, 44	opth 5340	. . . . . . 7 ⊢ (⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤)))
46	42, 45	bitrdi 290	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st ‘𝑧) = (1st ‘𝑤) ∧ (2nd ‘𝑧) = (2nd ‘𝑤))))
47	32, 40, 46	3bitr4d 314	. . . . 5 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ 𝑧 = 𝑤))
48	47	biimpd 232	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
49	48	rgen2 3132	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
50		dff13 7011	. . 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 710	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→ℂ
52		cnre 10689	. . . . . 6 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
53		oveq1 7163	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 + (i · 𝑦)) = (𝑢 + (i · 𝑦)))
54		oveq2 7164	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (i · 𝑦) = (i · 𝑣))
55	54	oveq2d 7172	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 + (i · 𝑦)) = (𝑢 + (i · 𝑣)))
56		ovex 7189	. . . . . . . . 9 ⊢ (𝑢 + (i · 𝑣)) ∈ V
57	53, 55, 1, 56	ovmpo 7311	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = (𝑢 + (i · 𝑣)))
58	57	eqeq2d 2769	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = (𝑢 + (i · 𝑣))))
59	58	2rexbiia 3222	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢 + (i · 𝑣)))
60	52, 59	sylibr 237	. . . . 5 ⊢ (𝑤 ∈ ℂ → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
61		fveq2 6663	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
62		df-ov 7159	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
63	61, 62	eqtr4di 2811	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
64	63	eqeq2d 2769	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
65	64	rexxp 5688	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
66	60, 65	sylibr 237	. . . 4 ⊢ (𝑤 ∈ ℂ → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
67	66	rgen 3080	. . 3 ⊢ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
68		dffo3 6865	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀𝑤 ∈ ℂ ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
69	26, 67, 68	mpbir2an 710	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→ℂ
70		df-f1o 6347	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
71	51, 69, 70	mpbir2an 710	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ⟨cop 4531   × cxp 5526   Fn wfn 6335  ⟶wf 6336  –1-1→wf1 6337  –onto→wfo 6338  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7697  2^nd c2nd 7698  ℂcc 10586  ℝcr 10587  ici 10590   + caddc 10591   · cmul 10593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349

This theorem is referenced by:  cnexALT  12439  cnrecnv  14585  cpnnen  15643  cnheiborlem  23669  mbfimaopnlem  24369