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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 4942 | . . 3 ⊢ (𝐽 qTop 𝐹) ⊆ 𝒫 ∪ (𝐽 qTop 𝐹) | |
2 | circtopn.j | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 24633 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2823 | . . . . 5 ⊢ 𝐽 ∈ Top |
5 | circtopn.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) | |
6 | circtopn.c | . . . . . 6 ⊢ 𝐶 = (◡abs “ {1}) | |
7 | 5, 6 | efifo 26436 | . . . . 5 ⊢ 𝐹:ℝ–onto→𝐶 |
8 | uniretop 24634 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
9 | 2 | unieqi 4914 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
10 | 8, 9 | eqtr4i 2757 | . . . . . 6 ⊢ ℝ = ∪ 𝐽 |
11 | 10 | qtopuni 23561 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:ℝ–onto→𝐶) → 𝐶 = ∪ (𝐽 qTop 𝐹)) |
12 | 4, 7, 11 | mp2an 689 | . . . 4 ⊢ 𝐶 = ∪ (𝐽 qTop 𝐹) |
13 | 12 | pweqi 4613 | . . 3 ⊢ 𝒫 𝐶 = 𝒫 ∪ (𝐽 qTop 𝐹) |
14 | 1, 13 | sseqtrri 4014 | . 2 ⊢ (𝐽 qTop 𝐹) ⊆ 𝒫 𝐶 |
15 | eqidd 2727 | . . . . 5 ⊢ (⊤ → (𝐹 “s ℝfld) = (𝐹 “s ℝfld)) | |
16 | rebase 21499 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ = (Base‘ℝfld)) |
18 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐹:ℝ–onto→𝐶) |
19 | recms 25263 | . . . . . 6 ⊢ ℝfld ∈ CMetSp | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝfld ∈ CMetSp) |
21 | 15, 17, 18, 20 | imasbas 17467 | . . . 4 ⊢ (⊤ → 𝐶 = (Base‘(𝐹 “s ℝfld))) |
22 | 21 | mptru 1540 | . . 3 ⊢ 𝐶 = (Base‘(𝐹 “s ℝfld)) |
23 | retopn 25262 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
24 | 2, 23 | eqtri 2754 | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℝfld) |
25 | eqid 2726 | . . . . . 6 ⊢ (TopSet‘(𝐹 “s ℝfld)) = (TopSet‘(𝐹 “s ℝfld)) | |
26 | 15, 17, 18, 20, 24, 25 | imastset 17477 | . . . . 5 ⊢ (⊤ → (TopSet‘(𝐹 “s ℝfld)) = (𝐽 qTop 𝐹)) |
27 | 26 | mptru 1540 | . . . 4 ⊢ (TopSet‘(𝐹 “s ℝfld)) = (𝐽 qTop 𝐹) |
28 | 27 | eqcomi 2735 | . . 3 ⊢ (𝐽 qTop 𝐹) = (TopSet‘(𝐹 “s ℝfld)) |
29 | 22, 28 | topnid 17390 | . 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 1533 ⊤wtru 1534 ∈ wcel 2098 ⊆ wss 3943 𝒫 cpw 4597 {csn 4623 ∪ cuni 4902 ↦ cmpt 5224 ◡ccnv 5668 ran crn 5670 “ cima 5672 –onto→wfo 6535 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 0cc0 11112 1c1 11113 ici 11114 · cmul 11117 2c2 12271 (,)cioo 13330 [,]cicc 13333 abscabs 15187 expce 16011 πcpi 16016 Basecbs 17153 TopSetcts 17212 TopOpenctopn 17376 topGenctg 17392 qTop cqtop 17458 “s cimas 17459 ℝfldcrefld 21497 Topctop 22750 CMetSpccms 25215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 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 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-refld 21498 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-fcls 23800 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-cfil 25138 df-cmet 25140 df-cms 25218 df-limc 25750 df-dv 25751 |
This theorem is referenced by: (None) |
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