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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > circcn | Structured version Visualization version GIF version |
Description: The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-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 |
---|---|
circcn | ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | circtopn.j | . . 3 ⊢ 𝐽 = (topGen‘ran (,)) | |
2 | retopon 22891 | . . 3 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | |
3 | 1, 2 | eqeltri 2872 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℝ) |
4 | circtopn.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) | |
5 | circtopn.c | . . . 4 ⊢ 𝐶 = (◡abs “ {1}) | |
6 | 4, 5 | efifo 24631 | . . 3 ⊢ 𝐹:ℝ–onto→𝐶 |
7 | fofn 6331 | . . 3 ⊢ (𝐹:ℝ–onto→𝐶 → 𝐹 Fn ℝ) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ 𝐹 Fn ℝ |
9 | qtopid 21833 | . 2 ⊢ ((𝐽 ∈ (TopOn‘ℝ) ∧ 𝐹 Fn ℝ) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) | |
10 | 3, 8, 9 | mp2an 684 | 1 ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 {csn 4366 ↦ cmpt 4920 ◡ccnv 5309 ran crn 5311 “ cima 5313 Fn wfn 6094 –onto→wfo 6097 ‘cfv 6099 (class class class)co 6876 ℝcr 10221 0cc0 10222 1c1 10223 ici 10224 · cmul 10227 2c2 11364 (,)cioo 12420 [,]cicc 12423 abscabs 14311 expce 15124 πcpi 15129 topGenctg 16409 qTop cqtop 16474 TopOnctopon 21039 Cn ccn 21353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-fi 8557 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-ioo 12424 df-ioc 12425 df-ico 12426 df-icc 12427 df-fz 12577 df-fzo 12717 df-fl 12844 df-mod 12920 df-seq 13052 df-exp 13111 df-fac 13310 df-bc 13339 df-hash 13367 df-shft 14144 df-cj 14176 df-re 14177 df-im 14178 df-sqrt 14312 df-abs 14313 df-limsup 14539 df-clim 14556 df-rlim 14557 df-sum 14754 df-ef 15130 df-sin 15132 df-cos 15133 df-pi 15135 df-struct 16182 df-ndx 16183 df-slot 16184 df-base 16186 df-sets 16187 df-ress 16188 df-plusg 16276 df-mulr 16277 df-starv 16278 df-sca 16279 df-vsca 16280 df-ip 16281 df-tset 16282 df-ple 16283 df-ds 16285 df-unif 16286 df-hom 16287 df-cco 16288 df-rest 16394 df-topn 16395 df-0g 16413 df-gsum 16414 df-topgen 16415 df-pt 16416 df-prds 16419 df-xrs 16473 df-qtop 16478 df-imas 16479 df-xps 16481 df-mre 16557 df-mrc 16558 df-acs 16560 df-mgm 17553 df-sgrp 17595 df-mnd 17606 df-submnd 17647 df-mulg 17853 df-cntz 18058 df-cmn 18506 df-psmet 20056 df-xmet 20057 df-met 20058 df-bl 20059 df-mopn 20060 df-fbas 20061 df-fg 20062 df-cnfld 20065 df-top 21023 df-topon 21040 df-topsp 21062 df-bases 21075 df-cld 21148 df-ntr 21149 df-cls 21150 df-nei 21227 df-lp 21265 df-perf 21266 df-cn 21356 df-cnp 21357 df-haus 21444 df-tx 21690 df-hmeo 21883 df-fil 21974 df-fm 22066 df-flim 22067 df-flf 22068 df-xms 22449 df-ms 22450 df-tms 22451 df-cncf 23005 df-limc 23967 df-dv 23968 |
This theorem is referenced by: (None) |
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