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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.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.) |
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 2827 | . . . . . 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 2758 | . . . . . . . 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 2841 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑥) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
15 | 9, 14 | sseldd 3982 | . . . . 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 2841 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn (𝐾 ↾t ℝ))) |
23 | 9, 22 | sseldd 3982 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ 𝑦) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
24 | 6 | mpomulcn 24605 | . . . . . . 7 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
26 | oveq12 7420 | . . . . . 6 ⊢ ((𝑢 = i ∧ 𝑣 = 𝑦) → (𝑢 · 𝑣) = (i · 𝑦)) | |
27 | 5, 5, 20, 23, 17, 17, 25, 26 | cnmpt22 23398 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (i · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
28 | 6 | addcn 24601 | . . . . . 6 ⊢ + ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
29 | 28 | a1i 11 | . . . . 5 ⊢ (⊤ → + ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
30 | 5, 5, 15, 27, 29 | cnmpt22f 23399 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
31 | 1, 30 | eqeltrid 2835 | . . 3 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) |
32 | 1 | cnrecnv 15116 | . . . 4 ⊢ ◡𝐹 = (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) |
33 | ref 15063 | . . . . . . . 8 ⊢ ℜ:ℂ⟶ℝ | |
34 | 33 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℜ:ℂ⟶ℝ) |
35 | 34 | feqmptd 6959 | . . . . . 6 ⊢ (⊤ → ℜ = (𝑧 ∈ ℂ ↦ (ℜ‘𝑧))) |
36 | recncf 24642 | . . . . . . 7 ⊢ ℜ ∈ (ℂ–cn→ℝ) | |
37 | ssid 4003 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
38 | ax-resscn 11169 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
39 | 16 | toponrestid 22643 | . . . . . . . . 9 ⊢ 𝐾 = (𝐾 ↾t ℂ) |
40 | 6, 39, 12 | cncfcn 24650 | . . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐾 Cn 𝐽)) |
41 | 37, 38, 40 | mp2an 688 | . . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐾 Cn 𝐽) |
42 | 36, 41 | eleqtri 2829 | . . . . . 6 ⊢ ℜ ∈ (𝐾 Cn 𝐽) |
43 | 35, 42 | eqeltrrdi 2840 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℜ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
44 | imf 15064 | . . . . . . . 8 ⊢ ℑ:ℂ⟶ℝ | |
45 | 44 | a1i 11 | . . . . . . 7 ⊢ (⊤ → ℑ:ℂ⟶ℝ) |
46 | 45 | feqmptd 6959 | . . . . . 6 ⊢ (⊤ → ℑ = (𝑧 ∈ ℂ ↦ (ℑ‘𝑧))) |
47 | imcncf 24643 | . . . . . . 7 ⊢ ℑ ∈ (ℂ–cn→ℝ) | |
48 | 47, 41 | eleqtri 2829 | . . . . . 6 ⊢ ℑ ∈ (𝐾 Cn 𝐽) |
49 | 46, 48 | eqeltrrdi 2840 | . . . . 5 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ (ℑ‘𝑧)) ∈ (𝐾 Cn 𝐽)) |
50 | 17, 43, 49 | cnmpt1t 23389 | . . . 4 ⊢ (⊤ → (𝑧 ∈ ℂ ↦ ⟨(ℜ‘𝑧), (ℑ‘𝑧)⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
51 | 32, 50 | eqeltrid 2835 | . . 3 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽))) |
52 | ishmeo 23483 | . . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) ↔ (𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn (𝐽 ×t 𝐽)))) | |
53 | 31, 51, 52 | sylanbrc 581 | . 2 ⊢ (⊤ → 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)) |
54 | 53 | mptru 1546 | 1 ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2104 ⊆ wss 3947 ⟨cop 4633 ↦ cmpt 5230 ◡ccnv 5674 ran crn 5676 ⟶wf 6538 ‘cfv 6542 (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 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 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 24700 mbfimaopnlem 25404 tpr2rico 33190 |
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