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Mirrors > Home > ILE Home > Th. List > cnrehmeocntop | GIF version |
Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 9443 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.) |
Ref | Expression |
---|---|
cnrehmeo.1 | ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) |
cnrehmeo.2 | ⊢ 𝐽 = (topGen‘ran (,)) |
cnrehmeocntop.3 | ⊢ 𝐾 = (MetOpen‘(abs ∘ − )) |
Ref | Expression |
---|---|
cnrehmeocntop | ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
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 12698 | . . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
4 | 2, 3 | eqeltri 2212 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
5 | 4 | a1i 9 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ)) |
6 | cnrehmeocntop.3 | . . . . . . . 8 ⊢ 𝐾 = (MetOpen‘(abs ∘ − )) | |
7 | 6 | cntoptop 12705 | . . . . . . 7 ⊢ 𝐾 ∈ Top |
8 | cnrest2r 12409 | . . . . . . 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 12460 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6 | tgioo2cntop 12721 | . . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
12 | 2, 11 | eqtri 2160 | . . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
13 | 12 | oveq2i 5785 | . . . . . . 7 ⊢ ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) |
14 | 10, 13 | eleqtrdi 2232 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
15 | 9, 14 | sseldd 3098 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
16 | 6 | cntoptopon 12704 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
17 | 16 | a1i 9 | . . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
18 | ax-icn 7718 | . . . . . . . 8 ⊢ i ∈ ℂ | |
19 | 18 | a1i 9 | . . . . . . 7 ⊢ (⊤ → i ∈ ℂ) |
20 | 5, 5, 17, 19 | cnmpt2c 12462 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
21 | 5, 5 | cnmpt2nd 12461 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
22 | 21, 13 | eleqtrdi 2232 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
23 | 9, 22 | sseldd 3098 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
24 | 6 | mulcncntop 12726 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
25 | 24 | a1i 9 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
26 | 5, 5, 20, 23, 25 | cnmpt22f 12467 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
27 | 6 | addcncntop 12724 | . . . . . 6 ⊢ + ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
28 | 27 | a1i 9 | . . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
29 | 5, 5, 15, 26, 28 | cnmpt22f 12467 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
30 | 1, 29 | eqeltrid 2226 | . . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
31 | 1 | cnrecnv 10685 | . . . 4 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
32 | ref 10630 | . . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ | |
33 | 32 | a1i 9 | . . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ) |
34 | 33 | feqmptd 5474 | . . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧))) |
35 | recncf 12745 | . . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ) | |
36 | ssid 3117 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
37 | ax-resscn 7715 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
38 | 16 | toponrestid 12191 | . . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾t ℂ) |
39 | 6, 38, 12 | cncfcncntop 12752 | . . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)) |
40 | 36, 37, 39 | mp2an 422 | . . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽) |
41 | 35, 40 | eleqtri 2214 | . . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽) |
42 | 34, 41 | eqeltrrdi 2231 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
43 | imf 10631 | . . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ | |
44 | 43 | a1i 9 | . . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ) |
45 | 44 | feqmptd 5474 | . . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧))) |
46 | imcncf 12746 | . . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ) | |
47 | 46, 40 | eleqtri 2214 | . . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽) |
48 | 45, 47 | eqeltrrdi 2231 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
49 | 17, 42, 48 | cnmpt1t 12457 | . . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
50 | 31, 49 | eqeltrid 2226 | . . 3 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
51 | ishmeo 12476 | . . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽)))) | |
52 | 30, 50, 51 | sylanbrc 413 | . 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)) |
53 | 52 | mptru 1340 | 1 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 ⊆ wss 3071 〈cop 3530 ↦ cmpt 3989 ◡ccnv 4538 ran crn 4540 ∘ ccom 4543 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 ℂcc 7621 ℝcr 7622 ici 7625 + caddc 7626 · cmul 7628 − cmin 7936 (,)cioo 9674 ℜcre 10615 ℑcim 10616 abscabs 10772 ↾t crest 12123 topGenctg 12138 MetOpencmopn 12157 Topctop 12167 TopOnctopon 12180 Cn ccn 12357 ×t ctx 12424 Homeochmeo 12472 –cn→ccncf 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 ax-addf 7745 ax-mulf 7746 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-sup 6871 df-inf 6872 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-xneg 9562 df-xadd 9563 df-ioo 9678 df-seqfrec 10222 df-exp 10296 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-rest 12125 df-topgen 12144 df-psmet 12159 df-xmet 12160 df-met 12161 df-bl 12162 df-mopn 12163 df-top 12168 df-topon 12181 df-bases 12213 df-cn 12360 df-cnp 12361 df-tx 12425 df-hmeo 12473 df-cncf 12730 |
This theorem is referenced by: (None) |
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