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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 4952 | . . 3 ⊢ (𝐽 qTop 𝐹) ⊆ 𝒫 ∪ (𝐽 qTop 𝐹) | |
2 | circtopn.j | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 24698 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2825 | . . . . 5 ⊢ 𝐽 ∈ Top |
5 | circtopn.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) | |
6 | circtopn.c | . . . . . 6 ⊢ 𝐶 = (◡abs “ {1}) | |
7 | 5, 6 | efifo 26501 | . . . . 5 ⊢ 𝐹:ℝ–onto→𝐶 |
8 | uniretop 24699 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
9 | 2 | unieqi 4924 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
10 | 8, 9 | eqtr4i 2759 | . . . . . 6 ⊢ ℝ = ∪ 𝐽 |
11 | 10 | qtopuni 23626 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:ℝ–onto→𝐶) → 𝐶 = ∪ (𝐽 qTop 𝐹)) |
12 | 4, 7, 11 | mp2an 690 | . . . 4 ⊢ 𝐶 = ∪ (𝐽 qTop 𝐹) |
13 | 12 | pweqi 4622 | . . 3 ⊢ 𝒫 𝐶 = 𝒫 ∪ (𝐽 qTop 𝐹) |
14 | 1, 13 | sseqtrri 4019 | . 2 ⊢ (𝐽 qTop 𝐹) ⊆ 𝒫 𝐶 |
15 | eqidd 2729 | . . . . 5 ⊢ (⊤ → (𝐹 “s ℝfld) = (𝐹 “s ℝfld)) | |
16 | rebase 21545 | . . . . . 6 ⊢ ℝ = (Base‘ℝfld) | |
17 | 16 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ = (Base‘ℝfld)) |
18 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐹:ℝ–onto→𝐶) |
19 | recms 25328 | . . . . . 6 ⊢ ℝfld ∈ CMetSp | |
20 | 19 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝfld ∈ CMetSp) |
21 | 15, 17, 18, 20 | imasbas 17501 | . . . 4 ⊢ (⊤ → 𝐶 = (Base‘(𝐹 “s ℝfld))) |
22 | 21 | mptru 1540 | . . 3 ⊢ 𝐶 = (Base‘(𝐹 “s ℝfld)) |
23 | retopn 25327 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (TopOpen‘ℝfld) | |
24 | 2, 23 | eqtri 2756 | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℝfld) |
25 | eqid 2728 | . . . . . 6 ⊢ (TopSet‘(𝐹 “s ℝfld)) = (TopSet‘(𝐹 “s ℝfld)) | |
26 | 15, 17, 18, 20, 24, 25 | imastset 17511 | . . . . 5 ⊢ (⊤ → (TopSet‘(𝐹 “s ℝfld)) = (𝐽 qTop 𝐹)) |
27 | 26 | mptru 1540 | . . . 4 ⊢ (TopSet‘(𝐹 “s ℝfld)) = (𝐽 qTop 𝐹) |
28 | 27 | eqcomi 2737 | . . 3 ⊢ (𝐽 qTop 𝐹) = (TopSet‘(𝐹 “s ℝfld)) |
29 | 22, 28 | topnid 17424 | . 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 3949 𝒫 cpw 4606 {csn 4632 ∪ cuni 4912 ↦ cmpt 5235 ◡ccnv 5681 ran crn 5683 “ cima 5685 –onto→wfo 6551 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 0cc0 11146 1c1 11147 ici 11148 · cmul 11151 2c2 12305 (,)cioo 13364 [,]cicc 13367 abscabs 15221 expce 16045 πcpi 16050 Basecbs 17187 TopSetcts 17246 TopOpenctopn 17410 topGenctg 17426 qTop cqtop 17492 “s cimas 17493 ℝfldcrefld 21543 Topctop 22815 CMetSpccms 25280 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-refld 21544 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-cmp 23311 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-fcls 23865 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-cfil 25203 df-cmet 25205 df-cms 25283 df-limc 25815 df-dv 25816 |
This theorem is referenced by: (None) |
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