Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cnrehmeo | Structured version Visualization version GIF version |
Description: The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12372 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 (,)) |
cnrehmeo.3 | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
cnrehmeo | ⊢ 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 23299 | . . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
4 | 2, 3 | eqeltri 2906 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
5 | 4 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ)) |
6 | cnrehmeo.3 | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 23319 | . . . . . . 7 ⊢ 𝐾 ∈ Top |
8 | cnrest2r 21823 | . . . . . . 7 ⊢ (𝐾 ∈ Top → ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) ⊆ ((𝐽 ×t 𝐽) Cn 𝐾)) | |
9 | 7, 8 | mp1i 13 | . . . . . 6 ⊢ (⊤ → ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) ⊆ ((𝐽 ×t 𝐽) Cn 𝐾)) |
10 | 5, 5 | cnmpt1st 22204 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6 | tgioo2 23338 | . . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
12 | 2, 11 | eqtri 2841 | . . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
13 | 12 | oveq2i 7156 | . . . . . . 7 ⊢ ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) |
14 | 10, 13 | eleqtrdi 2920 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
15 | 9, 14 | sseldd 3965 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
16 | 6 | cnfldtopon 23318 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
18 | ax-icn 10584 | . . . . . . . 8 ⊢ i ∈ ℂ | |
19 | 18 | a1i 11 | . . . . . . 7 ⊢ (⊤ → i ∈ ℂ) |
20 | 5, 5, 17, 19 | cnmpt2c 22206 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
21 | 5, 5 | cnmpt2nd 22205 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
22 | 21, 13 | eleqtrdi 2920 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
23 | 9, 22 | sseldd 3965 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
24 | 6 | mulcn 23402 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
26 | 5, 5, 20, 23, 25 | cnmpt22f 22211 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
27 | 6 | addcn 23400 | . . . . . 6 ⊢ + ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
28 | 27 | a1i 11 | . . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
29 | 5, 5, 15, 26, 28 | cnmpt22f 22211 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
30 | 1, 29 | eqeltrid 2914 | . . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
31 | 1 | cnrecnv 14512 | . . . 4 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) |
32 | ref 14459 | . . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ | |
33 | 32 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ) |
34 | 33 | feqmptd 6726 | . . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧))) |
35 | recncf 23437 | . . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ) | |
36 | ssid 3986 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
37 | ax-resscn 10582 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
38 | 16 | toponrestid 21457 | . . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾t ℂ) |
39 | 6, 38, 12 | cncfcn 23444 | . . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)) |
40 | 36, 37, 39 | mp2an 688 | . . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽) |
41 | 35, 40 | eleqtri 2908 | . . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽) |
42 | 34, 41 | syl6eqelr 2919 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
43 | imf 14460 | . . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ | |
44 | 43 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ) |
45 | 44 | feqmptd 6726 | . . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧))) |
46 | imcncf 23438 | . . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ) | |
47 | 46, 40 | eleqtri 2908 | . . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽) |
48 | 45, 47 | syl6eqelr 2919 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
49 | 17, 42, 48 | cnmpt1t 22201 | . . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ 〈(ℜ‘𝑧), (ℑ‘𝑧)〉) ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
50 | 31, 49 | eqeltrid 2914 | . . 3 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
51 | ishmeo 22295 | . . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽)))) | |
52 | 30, 50, 51 | sylanbrc 583 | . 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)) |
53 | 52 | mptru 1535 | 1 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 ⊆ wss 3933 〈cop 4563 ↦ cmpt 5137 ◡ccnv 5547 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ℂcc 10523 ℝcr 10524 ici 10527 + caddc 10528 · cmul 10530 (,)cioo 12726 ℜcre 14444 ℑcim 14445 ↾t crest 16682 TopOpenctopn 16683 topGenctg 16699 ℂfldccnfld 20473 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 ×t ctx 22096 Homeochmeo 22289 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 |
This theorem is referenced by: cnheiborlem 23485 mbfimaopnlem 24183 tpr2rico 31054 |
Copyright terms: Public domain | W3C validator |