Theorem cnref1o 8357

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 6717), 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		simpl 102	. . . . . . . 8 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → x ∈ ℝ)
2	1	recnd 6851	. . . . . . 7 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → x ∈ ℂ)
3		ax-icn 6778	. . . . . . . . 9 ⊢ i ∈ ℂ
4	3	a1i 9	. . . . . . . 8 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → i ∈ ℂ)
5		simpr 103	. . . . . . . . 9 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → y ∈ ℝ)
6	5	recnd 6851	. . . . . . . 8 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → y ∈ ℂ)
7	4, 6	mulcld 6845	. . . . . . 7 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → (i · y) ∈ ℂ)
8	2, 7	addcld 6844	. . . . . 6 ⊢ ((x ∈ ℝ ∧ y ∈ ℝ) → (x + (i · y)) ∈ ℂ)
9	8	rgen2a 2369	. . . . 5 ⊢ ∀x ∈ ℝ ∀y ∈ ℝ (x + (i · y)) ∈ ℂ
10		cnref1o.1	. . . . . 6 ⊢ 𝐹 = (x ∈ ℝ, y ∈ ℝ ↦ (x + (i · y)))
11	10	fnmpt2 5770	. . . . 5 ⊢ (∀x ∈ ℝ ∀y ∈ ℝ (x + (i · y)) ∈ ℂ → 𝐹 Fn (ℝ × ℝ))
12	9, 11	ax-mp 7	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
13		1st2nd2 5743	. . . . . . . . 9 ⊢ (z ∈ (ℝ × ℝ) → z = ⟨(1st ‘z), (2nd ‘z)⟩)
14	13	fveq2d 5125	. . . . . . . 8 ⊢ (z ∈ (ℝ × ℝ) → (𝐹‘z) = (𝐹‘⟨(1st ‘z), (2nd ‘z)⟩))
15		df-ov 5458	. . . . . . . 8 ⊢ ((1st ‘z)𝐹(2nd ‘z)) = (𝐹‘⟨(1st ‘z), (2nd ‘z)⟩)
16	14, 15	syl6eqr 2087	. . . . . . 7 ⊢ (z ∈ (ℝ × ℝ) → (𝐹‘z) = ((1st ‘z)𝐹(2nd ‘z)))
17		xp1st 5734	. . . . . . . 8 ⊢ (z ∈ (ℝ × ℝ) → (1st ‘z) ∈ ℝ)
18		xp2nd 5735	. . . . . . . 8 ⊢ (z ∈ (ℝ × ℝ) → (2nd ‘z) ∈ ℝ)
19	17	recnd 6851	. . . . . . . . 9 ⊢ (z ∈ (ℝ × ℝ) → (1st ‘z) ∈ ℂ)
20	3	a1i 9	. . . . . . . . . 10 ⊢ (z ∈ (ℝ × ℝ) → i ∈ ℂ)
21	18	recnd 6851	. . . . . . . . . 10 ⊢ (z ∈ (ℝ × ℝ) → (2nd ‘z) ∈ ℂ)
22	20, 21	mulcld 6845	. . . . . . . . 9 ⊢ (z ∈ (ℝ × ℝ) → (i · (2nd ‘z)) ∈ ℂ)
23	19, 22	addcld 6844	. . . . . . . 8 ⊢ (z ∈ (ℝ × ℝ) → ((1st ‘z) + (i · (2nd ‘z))) ∈ ℂ)
24		oveq1 5462	. . . . . . . . 9 ⊢ (x = (1st ‘z) → (x + (i · y)) = ((1st ‘z) + (i · y)))
25		oveq2 5463	. . . . . . . . . 10 ⊢ (y = (2nd ‘z) → (i · y) = (i · (2nd ‘z)))
26	25	oveq2d 5471	. . . . . . . . 9 ⊢ (y = (2nd ‘z) → ((1st ‘z) + (i · y)) = ((1st ‘z) + (i · (2nd ‘z))))
27	24, 26, 10	ovmpt2g 5577	. . . . . . . 8 ⊢ (((1st ‘z) ∈ ℝ ∧ (2nd ‘z) ∈ ℝ ∧ ((1st ‘z) + (i · (2nd ‘z))) ∈ ℂ) → ((1st ‘z)𝐹(2nd ‘z)) = ((1st ‘z) + (i · (2nd ‘z))))
28	17, 18, 23, 27	syl3anc 1134	. . . . . . 7 ⊢ (z ∈ (ℝ × ℝ) → ((1st ‘z)𝐹(2nd ‘z)) = ((1st ‘z) + (i · (2nd ‘z))))
29	16, 28	eqtrd 2069	. . . . . 6 ⊢ (z ∈ (ℝ × ℝ) → (𝐹‘z) = ((1st ‘z) + (i · (2nd ‘z))))
30	29, 23	eqeltrd 2111	. . . . 5 ⊢ (z ∈ (ℝ × ℝ) → (𝐹‘z) ∈ ℂ)
31	30	rgen 2368	. . . 4 ⊢ ∀z ∈ (ℝ × ℝ)(𝐹‘z) ∈ ℂ
32		ffnfv 5266	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶ℂ ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀z ∈ (ℝ × ℝ)(𝐹‘z) ∈ ℂ))
33	12, 31, 32	mpbir2an 848	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶ℂ
34	17, 18	jca 290	. . . . . . 7 ⊢ (z ∈ (ℝ × ℝ) → ((1st ‘z) ∈ ℝ ∧ (2nd ‘z) ∈ ℝ))
35		xp1st 5734	. . . . . . . 8 ⊢ (w ∈ (ℝ × ℝ) → (1st ‘w) ∈ ℝ)
36		xp2nd 5735	. . . . . . . 8 ⊢ (w ∈ (ℝ × ℝ) → (2nd ‘w) ∈ ℝ)
37	35, 36	jca 290	. . . . . . 7 ⊢ (w ∈ (ℝ × ℝ) → ((1st ‘w) ∈ ℝ ∧ (2nd ‘w) ∈ ℝ))
38		cru 7386	. . . . . . 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))))
39	34, 37, 38	syl2an 273	. . . . . 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))))
40		fveq2 5121	. . . . . . . . 9 ⊢ (z = w → (𝐹‘z) = (𝐹‘w))
41		fveq2 5121	. . . . . . . . . 10 ⊢ (z = w → (1st ‘z) = (1st ‘w))
42		fveq2 5121	. . . . . . . . . . 11 ⊢ (z = w → (2nd ‘z) = (2nd ‘w))
43	42	oveq2d 5471	. . . . . . . . . 10 ⊢ (z = w → (i · (2nd ‘z)) = (i · (2nd ‘w)))
44	41, 43	oveq12d 5473	. . . . . . . . 9 ⊢ (z = w → ((1st ‘z) + (i · (2nd ‘z))) = ((1st ‘w) + (i · (2nd ‘w))))
45	40, 44	eqeq12d 2051	. . . . . . . 8 ⊢ (z = w → ((𝐹‘z) = ((1st ‘z) + (i · (2nd ‘z))) ↔ (𝐹‘w) = ((1st ‘w) + (i · (2nd ‘w)))))
46	45, 29	vtoclga 2613	. . . . . . 7 ⊢ (w ∈ (ℝ × ℝ) → (𝐹‘w) = ((1st ‘w) + (i · (2nd ‘w))))
47	29, 46	eqeqan12d 2052	. . . . . 6 ⊢ ((z ∈ (ℝ × ℝ) ∧ w ∈ (ℝ × ℝ)) → ((𝐹‘z) = (𝐹‘w) ↔ ((1st ‘z) + (i · (2nd ‘z))) = ((1st ‘w) + (i · (2nd ‘w)))))
48		1st2nd2 5743	. . . . . . . 8 ⊢ (w ∈ (ℝ × ℝ) → w = ⟨(1st ‘w), (2nd ‘w)⟩)
49	13, 48	eqeqan12d 2052	. . . . . . 7 ⊢ ((z ∈ (ℝ × ℝ) ∧ w ∈ (ℝ × ℝ)) → (z = w ↔ ⟨(1st ‘z), (2nd ‘z)⟩ = ⟨(1st ‘w), (2nd ‘w)⟩))
50		vex 2554	. . . . . . . . 9 ⊢ z ∈ V
51		1stexg 5736	. . . . . . . . 9 ⊢ (z ∈ V → (1st ‘z) ∈ V)
52	50, 51	ax-mp 7	. . . . . . . 8 ⊢ (1st ‘z) ∈ V
53		2ndexg 5737	. . . . . . . . 9 ⊢ (z ∈ V → (2nd ‘z) ∈ V)
54	50, 53	ax-mp 7	. . . . . . . 8 ⊢ (2nd ‘z) ∈ V
55	52, 54	opth 3965	. . . . . . 7 ⊢ (⟨(1st ‘z), (2nd ‘z)⟩ = ⟨(1st ‘w), (2nd ‘w)⟩ ↔ ((1st ‘z) = (1st ‘w) ∧ (2nd ‘z) = (2nd ‘w)))
56	49, 55	syl6bb 185	. . . . . 6 ⊢ ((z ∈ (ℝ × ℝ) ∧ w ∈ (ℝ × ℝ)) → (z = w ↔ ((1st ‘z) = (1st ‘w) ∧ (2nd ‘z) = (2nd ‘w))))
57	39, 47, 56	3bitr4d 209	. . . . 5 ⊢ ((z ∈ (ℝ × ℝ) ∧ w ∈ (ℝ × ℝ)) → ((𝐹‘z) = (𝐹‘w) ↔ z = w))
58	57	biimpd 132	. . . 4 ⊢ ((z ∈ (ℝ × ℝ) ∧ w ∈ (ℝ × ℝ)) → ((𝐹‘z) = (𝐹‘w) → z = w))
59	58	rgen2a 2369	. . 3 ⊢ ∀z ∈ (ℝ × ℝ)∀w ∈ (ℝ × ℝ)((𝐹‘z) = (𝐹‘w) → z = w)
60		dff13 5350	. . 3 ⊢ (𝐹:(ℝ × ℝ)–1-1→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀z ∈ (ℝ × ℝ)∀w ∈ (ℝ × ℝ)((𝐹‘z) = (𝐹‘w) → z = w)))
61	33, 59, 60	mpbir2an 848	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→ℂ
62		cnre 6821	. . . . . 6 ⊢ (w ∈ ℂ → ∃u ∈ ℝ ∃v ∈ ℝ w = (u + (i · v)))
63		simpl 102	. . . . . . . . 9 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → u ∈ ℝ)
64		simpr 103	. . . . . . . . 9 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → v ∈ ℝ)
65	63	recnd 6851	. . . . . . . . . 10 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → u ∈ ℂ)
66	3	a1i 9	. . . . . . . . . . 11 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → i ∈ ℂ)
67	64	recnd 6851	. . . . . . . . . . 11 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → v ∈ ℂ)
68	66, 67	mulcld 6845	. . . . . . . . . 10 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → (i · v) ∈ ℂ)
69	65, 68	addcld 6844	. . . . . . . . 9 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → (u + (i · v)) ∈ ℂ)
70		oveq1 5462	. . . . . . . . . 10 ⊢ (x = u → (x + (i · y)) = (u + (i · y)))
71		oveq2 5463	. . . . . . . . . . 11 ⊢ (y = v → (i · y) = (i · v))
72	71	oveq2d 5471	. . . . . . . . . 10 ⊢ (y = v → (u + (i · y)) = (u + (i · v)))
73	70, 72, 10	ovmpt2g 5577	. . . . . . . . 9 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ ∧ (u + (i · v)) ∈ ℂ) → (u𝐹v) = (u + (i · v)))
74	63, 64, 69, 73	syl3anc 1134	. . . . . . . 8 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → (u𝐹v) = (u + (i · v)))
75	74	eqeq2d 2048	. . . . . . 7 ⊢ ((u ∈ ℝ ∧ v ∈ ℝ) → (w = (u𝐹v) ↔ w = (u + (i · v))))
76	75	2rexbiia 2334	. . . . . 6 ⊢ (∃u ∈ ℝ ∃v ∈ ℝ w = (u𝐹v) ↔ ∃u ∈ ℝ ∃v ∈ ℝ w = (u + (i · v)))
77	62, 76	sylibr 137	. . . . 5 ⊢ (w ∈ ℂ → ∃u ∈ ℝ ∃v ∈ ℝ w = (u𝐹v))
78		fveq2 5121	. . . . . . . 8 ⊢ (z = ⟨u, v⟩ → (𝐹‘z) = (𝐹‘⟨u, v⟩))
79		df-ov 5458	. . . . . . . 8 ⊢ (u𝐹v) = (𝐹‘⟨u, v⟩)
80	78, 79	syl6eqr 2087	. . . . . . 7 ⊢ (z = ⟨u, v⟩ → (𝐹‘z) = (u𝐹v))
81	80	eqeq2d 2048	. . . . . 6 ⊢ (z = ⟨u, v⟩ → (w = (𝐹‘z) ↔ w = (u𝐹v)))
82	81	rexxp 4423	. . . . 5 ⊢ (∃z ∈ (ℝ × ℝ)w = (𝐹‘z) ↔ ∃u ∈ ℝ ∃v ∈ ℝ w = (u𝐹v))
83	77, 82	sylibr 137	. . . 4 ⊢ (w ∈ ℂ → ∃z ∈ (ℝ × ℝ)w = (𝐹‘z))
84	83	rgen 2368	. . 3 ⊢ ∀w ∈ ℂ ∃z ∈ (ℝ × ℝ)w = (𝐹‘z)
85		dffo3 5257	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→ℂ ↔ (𝐹:(ℝ × ℝ)⟶ℂ ∧ ∀w ∈ ℂ ∃z ∈ (ℝ × ℝ)w = (𝐹‘z)))
86	33, 84, 85	mpbir2an 848	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→ℂ
87		df-f1o 4852	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→ℂ ↔ (𝐹:(ℝ × ℝ)–1-1→ℂ ∧ 𝐹:(ℝ × ℝ)–onto→ℂ))
88	61, 86, 87	mpbir2an 848	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→ℂ

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  Vcvv 2551  ⟨cop 3370   × cxp 4286   Fn wfn 4840  ⟶wf 4841  –1-1→wf1 4842  –onto→wfo 4843  –1-1-onto→wf1o 4844  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1^st c1st 5707  2^nd c2nd 5708  ℂcc 6709  ℝcr 6710  ici 6713   + caddc 6714   · cmul 6716

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-ltxr 6862  df-sub 6981  df-neg 6982  df-reap 7359

This theorem is referenced by:  cnrecnv  9138