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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 12973 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 24500 | . . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
4 | 2, 3 | eqeltri 2829 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
5 | 4 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℝ)) |
6 | cnrehmeo.3 | . . . . . . . 8 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
7 | 6 | cnfldtop 24520 | . . . . . . 7 ⊢ 𝐾 ∈ Top |
8 | cnrest2r 23011 | . . . . . . 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 23392 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
11 | 6 | tgioo2 24539 | . . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
12 | 2, 11 | eqtri 2760 | . . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾t ℝ) |
13 | 12 | oveq2i 7422 | . . . . . . 7 ⊢ ((𝐽 ×t 𝐽) Cn 𝐽) = ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ)) |
14 | 10, 13 | eleqtrdi 2843 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
15 | 9, 14 | sseldd 3983 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
16 | 6 | cnfldtopon 24519 | . . . . . . . 8 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 𝐾 ∈ (TopOn‘ℂ)) |
18 | ax-icn 11171 | . . . . . . . 8 ⊢ i ∈ ℂ | |
19 | 18 | a1i 11 | . . . . . . 7 ⊢ (⊤ → i ∈ ℂ) |
20 | 5, 5, 17, 19 | cnmpt2c 23394 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ i) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
21 | 5, 5 | cnmpt2nd 23393 | . . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
22 | 21, 13 | eleqtrdi 2843 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
23 | 9, 22 | sseldd 3983 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
24 | 6 | mulcn 24603 | . . . . . . 7 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (⊤ → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
26 | 5, 5, 20, 23, 25 | cnmpt22f 23399 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
27 | 6 | addcn 24601 | . . . . . 6 ⊢ + ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
28 | 27 | a1i 11 | . . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
29 | 5, 5, 15, 26, 28 | cnmpt22f 23399 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
30 | 1, 29 | eqeltrid 2837 | . . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
31 | 1 | cnrecnv 15116 | . . . 4 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
32 | ref 15063 | . . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ | |
33 | 32 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ) |
34 | 33 | feqmptd 6960 | . . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧))) |
35 | recncf 24642 | . . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ) | |
36 | ssid 4004 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
37 | ax-resscn 11169 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
38 | 16 | toponrestid 22643 | . . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾t ℂ) |
39 | 6, 38, 12 | cncfcn 24650 | . . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)) |
40 | 36, 37, 39 | mp2an 690 | . . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽) |
41 | 35, 40 | eleqtri 2831 | . . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽) |
42 | 34, 41 | eqeltrrdi 2842 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
43 | imf 15064 | . . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ | |
44 | 43 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ) |
45 | 44 | feqmptd 6960 | . . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧))) |
46 | imcncf 24643 | . . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ) | |
47 | 46, 40 | eleqtri 2831 | . . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽) |
48 | 45, 47 | eqeltrrdi 2842 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
49 | 17, 42, 48 | cnmpt1t 23389 | . . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
50 | 31, 49 | eqeltrid 2837 | . . 3 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
51 | ishmeo 23483 | . . 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 1548 | 1 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ⊆ wss 3948 ⟨cop 4634 ↦ cmpt 5231 ◡ccnv 5675 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 ℂcc 11110 ℝcr 11111 ici 11114 + caddc 11115 · cmul 11117 (,)cioo 13328 ℜcre 15048 ℑcim 15049 ↾t crest 17370 TopOpenctopn 17371 topGenctg 17387 ℂfldccnfld 21144 Topctop 22615 TopOnctopon 22632 Cn ccn 22948 ×t ctx 23284 Homeochmeo 23477 –cn→ccncf 24616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-icc 13335 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18987 df-cntz 19222 df-cmn 19691 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cn 22951 df-cnp 22952 df-tx 23286 df-hmeo 23479 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 |
This theorem is referenced by: cnheiborlem 24694 mbfimaopnlem 25396 tpr2rico 33178 |
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