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Theorem cnrehmeocntop 15353

Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 9885 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 15269	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
4	2, 3	eqeltri 2304	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ)
5	4	a1i 9	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ))
6		cnrehmeocntop.3	. . . . . . . 8 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
7	6	cntoptop 15276	. . . . . . 7 ⊢ 𝐾 ∈ Top
8		cnrest2r 14980	. . . . . . 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 15031	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
11	6	tgioo2cntop 15300	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
12	2, 11	eqtri 2252	. . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
13	12	oveq2i 6029	. . . . . . 7 ⊢ ((𝐽 ×_t 𝐽) Cn 𝐽) = ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ))
14	10, 13	eleqtrdi 2324	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
15	9, 14	sseldd 3228	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
16	6	cntoptopon 15275	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
17	16	a1i 9	. . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
18		ax-icn 8127	. . . . . . . 8 ⊢ i ∈ ℂ
19	18	a1i 9	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
20	5, 5, 17, 19	cnmpt2c 15033	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
21	5, 5	cnmpt2nd 15032	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
22	21, 13	eleqtrdi 2324	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn (𝐾 ↾_t ℝ)))
23	9, 22	sseldd 3228	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
24	6	mulcncntop 15307	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
25	24	a1i 9	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
26	5, 5, 20, 23, 25	cnmpt22f 15038	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
27	6	addcncntop 15305	. . . . . 6 ⊢ + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
28	27	a1i 9	. . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
29	5, 5, 15, 26, 28	cnmpt22f 15038	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
30	1, 29	eqeltrid 2318	. . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
31	1	cnrecnv 11488	. . . 4 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
32		ref 11433	. . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ
33	32	a1i 9	. . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ)
34	33	feqmptd 5699	. . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)))
35		recncf 15329	. . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ)
36		ssid 3247	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
37		ax-resscn 8124	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
38	16	toponrestid 14764	. . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾_t ℂ)
39	6, 38, 12	cncfcncntop 15336	. . . . . . . 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 2306	. . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽)
42	34, 41	eqeltrrdi 2323	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽))
43		imf 11434	. . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ
44	43	a1i 9	. . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ)
45	44	feqmptd 5699	. . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)))
46		imcncf 15330	. . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ)
47	46, 40	eleqtri 2306	. . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽)
48	45, 47	eqeltrrdi 2323	. . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽))
49	17, 42, 48	cnmpt1t 15028	. . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
50	31, 49	eqeltrid 2318	. . 3 ⊢ (⊤ → ^◡𝐹 ∈ (𝐾 Cn (𝐽 ×_t 𝐽)))
51		ishmeo 15047	. . 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 1406	1 ⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Colors of variables: wff set class

Syntax hints: = wceq 1397 ⊤wtru 1398 ∈ wcel 2202 ⊆ wss 3200 ⟨cop 3672 ↦ cmpt 4150 ^◡ccnv 4724 ran crn 4726 ∘ ccom 4729 ⟶wf 5322 ‘cfv 5326 (class class class)co 6018 ∈ cmpo 6020 ℂcc 8030 ℝcr 8031 ici 8034 + caddc 8035 · cmul 8037 − cmin 8350 (,)cioo 10123 ℜcre 11418 ℑcim 11419 abscabs 11575 ↾_t crest 13340 topGenctg 13355 MetOpencmopn 14574 Topctop 14740 TopOnctopon 14753 Cn ccn 14928 ×_t ctx 14995 Homeochmeo 15043 –cn→ccncf 15313

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-map 6819 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-seqfrec 10711 df-exp 10802 df-cj 11420 df-re 11421 df-im 11422 df-rsqrt 11576 df-abs 11577 df-rest 13342 df-topgen 13361 df-psmet 14576 df-xmet 14577 df-met 14578 df-bl 14579 df-mopn 14580 df-top 14741 df-topon 14754 df-bases 14786 df-cn 14931 df-cnp 14932 df-tx 14996 df-hmeo 15044 df-cncf 15314

This theorem is referenced by: (None)

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