Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > circtopn | Structured version Visualization version GIF version |
Description: The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.) |
Ref | Expression |
---|---|
circtopn.i | ⊢ 𝐼 = (0[,](2 · π)) |
circtopn.j | ⊢ 𝐽 = (topGen‘ran (,)) |
circtopn.f | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) |
circtopn.c | ⊢ 𝐶 = (◡abs “ {1}) |
Ref | Expression |
---|---|
circtopn | ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4474 | . . 3 ⊢ (𝐽 qTop 𝐹) ⊆ 𝒫 ∪ (𝐽 qTop 𝐹) | |
2 | circtopn.j | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 22565 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2697 | . . . . 5 ⊢ 𝐽 ∈ Top |
5 | circtopn.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) | |
6 | circtopn.c | . . . . . 6 ⊢ 𝐶 = (◡abs “ {1}) | |
7 | 5, 6 | efifo 24293 | . . . . 5 ⊢ 𝐹:ℝ–onto→𝐶 |
8 | uniretop 22566 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
9 | 2 | unieqi 4445 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
10 | 8, 9 | eqtr4i 2647 | . . . . . 6 ⊢ ℝ = ∪ 𝐽 |
11 | 10 | qtopuni 21505 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:ℝ–onto→𝐶) → 𝐶 = ∪ (𝐽 qTop 𝐹)) |
12 | 4, 7, 11 | mp2an 708 | . . . 4 ⊢ 𝐶 = ∪ (𝐽 qTop 𝐹) |
13 | 12 | pweqi 4162 | . . 3 ⊢ 𝒫 𝐶 = 𝒫 ∪ (𝐽 qTop 𝐹) |
14 | 1, 13 | sseqtr4i 3638 | . 2 ⊢ (𝐽 qTop 𝐹) ⊆ 𝒫 𝐶 |
15 | eqidd 2623 | . . . . 5 ⊢ (⊤ → (𝐹 “s ℝfld) = (𝐹 “s ℝfld)) | |
16 | rebase 19952 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ = (Base‘ℝfld)) |
18 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐹:ℝ–onto→𝐶) |
19 | recms 23168 | . . . . . 6 ⊢ ℝfld ∈ CMetSp | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝfld ∈ CMetSp) |
21 | 15, 17, 18, 20 | imasbas 16172 | . . . 4 ⊢ (⊤ → 𝐶 = (Base‘(𝐹 “s ℝfld))) |
22 | 21 | trud 1493 | . . 3 ⊢ 𝐶 = (Base‘(𝐹 “s ℝfld)) |
23 | retopn 23167 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
24 | 2, 23 | eqtri 2644 | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℝfld) |
25 | eqid 2622 | . . . . . 6 ⊢ (TopSet‘(𝐹 “s ℝfld)) = (TopSet‘(𝐹 “s ℝfld)) | |
26 | 15, 17, 18, 20, 24, 25 | imastset 16182 | . . . . 5 ⊢ (⊤ → (TopSet‘(𝐹 “s ℝfld)) = (𝐽 qTop 𝐹)) |
27 | 26 | trud 1493 | . . . 4 ⊢ (TopSet‘(𝐹 “s ℝfld)) = (𝐽 qTop 𝐹) |
28 | 27 | eqcomi 2631 | . . 3 ⊢ (𝐽 qTop 𝐹) = (TopSet‘(𝐹 “s ℝfld)) |
29 | 22, 28 | topnid 16096 | . 2 ⊢ ((𝐽 qTop 𝐹) ⊆ 𝒫 𝐶 → (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld))) |
30 | 14, 29 | ax-mp 5 | 1 ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 ↦ cmpt 4729 ◡ccnv 5113 ran crn 5115 “ cima 5117 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 ici 9938 · cmul 9941 2c2 11070 (,)cioo 12175 [,]cicc 12178 abscabs 13974 expce 14792 πcpi 14797 Basecbs 15857 TopSetcts 15947 TopOpenctopn 16082 topGenctg 16098 qTop cqtop 16163 “s cimas 16164 ℝfldcrefld 19950 Topctop 20698 CMetSpccms 23129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-fcls 21745 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-cfil 23053 df-cmet 23055 df-cms 23132 df-limc 23630 df-dv 23631 |
This theorem is referenced by: (None) |
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