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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 9469 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 12734 | . . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
4 | 2, 3 | eqeltri 2213 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
5 | 4 | a1i 9 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ)) |
6 | cnrehmeocntop.3 | . . . . . . . 8 ⊢ 𝐾 = (MetOpen‘(abs ∘ − )) | |
7 | 6 | cntoptop 12741 | . . . . . . 7 ⊢ 𝐾 ∈ Top |
8 | cnrest2r 12445 | . . . . . . 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 12496 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6 | tgioo2cntop 12757 | . . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
12 | 2, 11 | eqtri 2161 | . . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
13 | 12 | oveq2i 5793 | . . . . . . 7 ⊢ ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) |
14 | 10, 13 | eleqtrdi 2233 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
15 | 9, 14 | sseldd 3103 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
16 | 6 | cntoptopon 12740 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
17 | 16 | a1i 9 | . . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
18 | ax-icn 7739 | . . . . . . . 8 ⊢ i ∈ ℂ | |
19 | 18 | a1i 9 | . . . . . . 7 ⊢ (⊤ → i ∈ ℂ) |
20 | 5, 5, 17, 19 | cnmpt2c 12498 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
21 | 5, 5 | cnmpt2nd 12497 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
22 | 21, 13 | eleqtrdi 2233 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
23 | 9, 22 | sseldd 3103 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
24 | 6 | mulcncntop 12762 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
25 | 24 | a1i 9 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
26 | 5, 5, 20, 23, 25 | cnmpt22f 12503 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
27 | 6 | addcncntop 12760 | . . . . . 6 ⊢ + ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
28 | 27 | a1i 9 | . . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
29 | 5, 5, 15, 26, 28 | cnmpt22f 12503 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
30 | 1, 29 | eqeltrid 2227 | . . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
31 | 1 | cnrecnv 10714 | . . . 4 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
32 | ref 10659 | . . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ | |
33 | 32 | a1i 9 | . . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ) |
34 | 33 | feqmptd 5482 | . . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧))) |
35 | recncf 12781 | . . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ) | |
36 | ssid 3122 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
37 | ax-resscn 7736 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
38 | 16 | toponrestid 12227 | . . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾t ℂ) |
39 | 6, 38, 12 | cncfcncntop 12788 | . . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)) |
40 | 36, 37, 39 | mp2an 423 | . . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽) |
41 | 35, 40 | eleqtri 2215 | . . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽) |
42 | 34, 41 | eqeltrrdi 2232 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
43 | imf 10660 | . . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ | |
44 | 43 | a1i 9 | . . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ) |
45 | 44 | feqmptd 5482 | . . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧))) |
46 | imcncf 12782 | . . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ) | |
47 | 46, 40 | eleqtri 2215 | . . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽) |
48 | 45, 47 | eqeltrrdi 2232 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
49 | 17, 42, 48 | cnmpt1t 12493 | . . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
50 | 31, 49 | eqeltrid 2227 | . . 3 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
51 | ishmeo 12512 | . . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽)))) | |
52 | 30, 50, 51 | sylanbrc 414 | . 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)) |
53 | 52 | mptru 1341 | 1 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ⊤wtru 1333 ∈ wcel 1481 ⊆ wss 3076 〈cop 3535 ↦ cmpt 3997 ◡ccnv 4546 ran crn 4548 ∘ ccom 4551 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 ℂcc 7642 ℝcr 7643 ici 7646 + caddc 7647 · cmul 7649 − cmin 7957 (,)cioo 9701 ℜcre 10644 ℑcim 10645 abscabs 10801 ↾t crest 12159 topGenctg 12174 MetOpencmopn 12193 Topctop 12203 TopOnctopon 12216 Cn ccn 12393 ×t ctx 12460 Homeochmeo 12508 –cn→ccncf 12765 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-addf 7766 ax-mulf 7767 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-map 6552 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-cn 12396 df-cnp 12397 df-tx 12461 df-hmeo 12509 df-cncf 12766 |
This theorem is referenced by: (None) |
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