cnrehmeocntop - Intuitionistic Logic Explorer

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Theorem cnrehmeocntop 15421

Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 9946 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 15337	. . . . . . 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 15344	. . . . . . 7 ⊢ 𝐾 ∈ Top
8		cnrest2r 15048	. . . . . . 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 15099	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
11	6	tgioo2cntop 15368	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾_t ℝ)
12	2, 11	eqtri 2252	. . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾_t ℝ)
13	12	oveq2i 6039	. . . . . . 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 3229	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
16	6	cntoptopon 15343	. . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ)
17	16	a1i 9	. . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ))
18		ax-icn 8187	. . . . . . . 8 ⊢ i ∈ ℂ
19	18	a1i 9	. . . . . . 7 ⊢ (⊤ → i ∈ ℂ)
20	5, 5, 17, 19	cnmpt2c 15101	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
21	5, 5	cnmpt2nd 15100	. . . . . . . 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 3229	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
24	6	mulcncntop 15375	. . . . . . 7 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
25	24	a1i 9	. . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
26	5, 5, 20, 23, 25	cnmpt22f 15106	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐾))
27	6	addcncntop 15373	. . . . . 6 ⊢ + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
28	27	a1i 9	. . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
29	5, 5, 15, 26, 28	cnmpt22f 15106	. . . 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 11550	. . . 4 ⊢ ^◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩)
32		ref 11495	. . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ
33	32	a1i 9	. . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ)
34	33	feqmptd 5708	. . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)))
35		recncf 15397	. . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ)
36		ssid 3248	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
37		ax-resscn 8184	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
38	16	toponrestid 14832	. . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾_t ℂ)
39	6, 38, 12	cncfcncntop 15404	. . . . . . . 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 11496	. . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ
44	43	a1i 9	. . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ)
45	44	feqmptd 5708	. . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)))
46		imcncf 15398	. . . . . . 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 15096	. . . 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 15115	. . 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 1407	1 ⊢ 𝐹 ∈ ((𝐽 ×_t 𝐽)Homeo𝐾)

Colors of variables: wff set class

Syntax hints: = wceq 1398 ⊤wtru 1399 ∈ wcel 2202 ⊆ wss 3201 ⟨cop 3676 ↦ cmpt 4155 ^◡ccnv 4730 ran crn 4732 ∘ ccom 4735 ⟶wf 5329 ‘cfv 5333 (class class class)co 6028 ∈ cmpo 6030 ℂcc 8090 ℝcr 8091 ici 8094 + caddc 8095 · cmul 8097 − cmin 8409 (,)cioo 10184 ℜcre 11480 ℑcim 11481 abscabs 11637 ↾_t crest 13402 topGenctg 13417 MetOpencmopn 14637 Topctop 14808 TopOnctopon 14821 Cn ccn 14996 ×_t ctx 15063 Homeochmeo 15111 –cn→ccncf 15381

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210 ax-arch 8211 ax-caucvg 8212 ax-addf 8214 ax-mulf 8215

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-map 6862 df-sup 7243 df-inf 7244 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-inn 9203 df-2 9261 df-3 9262 df-4 9263 df-n0 9462 df-z 9541 df-uz 9817 df-q 9915 df-rp 9950 df-xneg 10068 df-xadd 10069 df-ioo 10188 df-seqfrec 10773 df-exp 10864 df-cj 11482 df-re 11483 df-im 11484 df-rsqrt 11638 df-abs 11639 df-rest 13404 df-topgen 13423 df-psmet 14639 df-xmet 14640 df-met 14641 df-bl 14642 df-mopn 14643 df-top 14809 df-topon 14822 df-bases 14854 df-cn 14999 df-cnp 15000 df-tx 15064 df-hmeo 15112 df-cncf 15382

This theorem is referenced by: (None)

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