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Theorem cnrehmeocntop 14846

Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 9725 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)

**Hypotheses**
Ref	Expression
cnrehmeo.1	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
cnrehmeo.2	⊢ 𝐽 = (topGen‘ran (,))
cnrehmeocntop.3	⊢ 𝐾 = (MetOpen‘(abs ∘ − ))

**Assertion**
Ref	Expression
cnrehmeocntop	⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Distinct variable group: 𝑥,𝑦,𝐾 Allowed substitution hints: 𝐹(𝑥,𝑦) 𝐽(𝑥,𝑦)

Proof of Theorem cnrehmeocntop Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnrehmeo.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))
2		cnrehmeo.2	. . . . . . 7 ⊢ 𝐽 = (topGen‘ran (,))
3		retopon 14762	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
4	2, 3	eqeltri 2269	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ)
5	4	a1i 9	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ))
6		cnrehmeocntop.3	. . . . . . . 8 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
7	6	cntoptop 14769	. . . . . . 7 ⊢ 𝐾 ∈ Top
8		cnrest2r 14473	. . . . . . 7 ⊢ (𝐾 ∈ Top → ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)) ⊆ ((𝐽 ×_t 𝐽) Cn 𝐾))
9	7, 8	mp1i 10	. . . . . 6 ⊢ (⊤ → ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)) ⊆ ((𝐽 ×_t 𝐽) Cn 𝐾))
10	5, 5	cnmpt1st 14524	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
11	6	tgioo2cntop 14793	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
12	2, 11	eqtri 2217	. . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
13	12	oveq2i 5933	. . . . . . 7 ⊢ ((𝐽 ×_t 𝐽) Cn 𝐽) = ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ))
14	10, 13	eleqtrdi 2289	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
15	9, 14	sseldd 3184	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
16	6	cntoptopon 14768	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
17	16	a1i 9	. . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
18		ax-icn 7974	. . . . . . . 8 ⊢ i ∈ ℂ
19	18	a1i 9	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
20	5, 5, 17, 19	cnmpt2c 14526	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
21	5, 5	cnmpt2nd 14525	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	21, 13	eleqtrdi 2289	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
23	9, 22	sseldd 3184	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
24	6	mulcncntop 14800	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
25	24	a1i 9	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
26	5, 5, 20, 23, 25	cnmpt22f 14531	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
27	6	addcncntop 14798	. . . . . 6 ⊢ + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
28	27	a1i 9	. . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
29	5, 5, 15, 26, 28	cnmpt22f 14531	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
30	1, 29	eqeltrid 2283	. . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
31	1	cnrecnv 11075	. . . 4 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
32		ref 11020	. . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ
33	32	a1i 9	. . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ)
34	33	feqmptd 5614	. . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)))
35		recncf 14822	. . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ)
36		ssid 3203	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
37		ax-resscn 7971	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
38	16	toponrestid 14257	. . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾_t ℂ)
39	6, 38, 12	cncfcncntop 14829	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽))
40	36, 37, 39	mp2an 426	. . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)
41	35, 40	eleqtri 2271	. . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽)
42	34, 41	eqeltrrdi 2288	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽))
43		imf 11021	. . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ
44	43	a1i 9	. . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ)
45	44	feqmptd 5614	. . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)))
46		imcncf 14823	. . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ)
47	46, 40	eleqtri 2271	. . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽)
48	45, 47	eqeltrrdi 2288	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽))
49	17, 42, 48	cnmpt1t 14521	. . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
50	31, 49	eqeltrid 2283	. . 3 ⊢ (⊤ → ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
51		ishmeo 14540	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾) ∧ ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽))))
52	30, 50, 51	sylanbrc 417	. 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾))
53	52	mptru 1373	1 ⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Colors of variables: wff set class

Syntax hints: = wceq 1364 ⊤wtru 1365 ∈ wcel 2167 ⊆ wss 3157 ⟨cop 3625 ↦ cmpt 4094 ^◡ccnv 4662 ran crn 4664 ∘ ccom 4667 ⟶wf 5254 ‘cfv 5258 (class class class)co 5922 ∈ cmpo 5924 ℂcc 7877 ℝcr 7878 ici 7881 + caddc 7882 · cmul 7884 − cmin 8197 (,)cioo 9963 ℜcre 11005 ℑcim 11006 abscabs 11162 ↾_t crest 12910 topGenctg 12925 MetOpencmopn 14097 Topctop 14233 TopOnctopon 14246 Cn ccn 14421 ×_t ctx 14488 Homeochmeo 14536 –cn→ccncf 14806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999 ax-addf 8001 ax-mulf 8002

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-map 6709 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-xneg 9847 df-xadd 9848 df-ioo 9967 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-rest 12912 df-topgen 12931 df-psmet 14099 df-xmet 14100 df-met 14101 df-bl 14102 df-mopn 14103 df-top 14234 df-topon 14247 df-bases 14279 df-cn 14424 df-cnp 14425 df-tx 14489 df-hmeo 14537 df-cncf 14807

This theorem is referenced by: (None)

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