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Mirrors > Home > MPE Home > Th. List > cnfldtopon | Structured version Visualization version GIF version |
Description: The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
cnfldtopn.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
cnfldtopon | ⊢ 𝐽 ∈ (TopOn‘ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldtps 22951 | . 2 ⊢ ℂfld ∈ TopSp | |
2 | cnfldbas 20110 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
3 | cnfldtopn.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
4 | 2, 3 | istps 21109 | . 2 ⊢ (ℂfld ∈ TopSp ↔ 𝐽 ∈ (TopOn‘ℂ)) |
5 | 1, 4 | mpbi 222 | 1 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∈ wcel 2166 ‘cfv 6123 ℂcc 10250 TopOpenctopn 16435 ℂfldccnfld 20106 TopOnctopon 21085 TopSpctps 21107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-fz 12620 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-rest 16436 df-topn 16437 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-xms 22495 df-ms 22496 |
This theorem is referenced by: cnfldtop 22957 unicntop 22959 sszcld 22990 reperflem 22991 cnperf 22993 divcn 23041 fsumcn 23043 expcn 23045 divccn 23046 cncfcn1 23083 cncfmptc 23084 cncfmptid 23085 cncfmpt2f 23087 cdivcncf 23090 abscncfALT 23093 cncfcnvcn 23094 cnmptre 23096 iirevcn 23099 iihalf1cn 23101 iihalf2cn 23103 iimulcn 23107 icchmeo 23110 cnrehmeo 23122 cnheiborlem 23123 cnheibor 23124 cnllycmp 23125 evth 23128 evth2 23129 lebnumlem2 23131 reparphti 23166 pcoass 23193 csscld 23417 clsocv 23418 cncmet 23490 resscdrg 23526 mbfimaopnlem 23821 limcvallem 24034 ellimc2 24040 limcnlp 24041 limcflflem 24043 limcflf 24044 limcmo 24045 limcres 24049 cnplimc 24050 cnlimc 24051 limccnp 24054 limccnp2 24055 limciun 24057 dvbss 24064 perfdvf 24066 recnperf 24068 dvreslem 24072 dvres2lem 24073 dvres3a 24077 dvidlem 24078 dvcnp2 24082 dvcn 24083 dvnres 24093 dvaddbr 24100 dvmulbr 24101 dvcmulf 24107 dvcobr 24108 dvcjbr 24111 dvrec 24117 dvmptid 24119 dvmptc 24120 dvmptres2 24124 dvmptcmul 24126 dvmptntr 24133 dvmptfsum 24137 dvcnvlem 24138 dvcnv 24139 dvexp3 24140 dveflem 24141 dvlipcn 24156 lhop1lem 24175 lhop2 24177 lhop 24178 dvcnvrelem2 24180 dvcnvre 24181 ftc1lem3 24200 ftc1cn 24205 plycn 24416 dvply1 24438 dvtaylp 24523 taylthlem1 24526 taylthlem2 24527 ulmdvlem3 24555 psercn2 24576 psercn 24579 pserdvlem2 24581 pserdv 24582 abelth 24594 pige3 24669 logcn 24792 dvloglem 24793 logdmopn 24794 dvlog 24796 dvlog2 24798 efopnlem2 24802 efopn 24803 logtayl 24805 dvcxp1 24883 cxpcn 24888 cxpcn2 24889 cxpcn3 24891 resqrtcn 24892 sqrtcn 24893 loglesqrt 24901 atansopn 25072 dvatan 25075 xrlimcnp 25108 efrlim 25109 lgamucov 25177 lgamucov2 25178 ftalem3 25214 vmcn 28109 dipcn 28130 ipasslem7 28246 ipasslem8 28247 occllem 28717 nlelchi 29475 tpr2rico 30503 rmulccn 30519 raddcn 30520 cxpcncf1 31222 cvxpconn 31770 cvxsconn 31771 cnllysconn 31773 sinccvglem 32110 ivthALT 32868 knoppcnlem10 33025 knoppcnlem11 33026 broucube 33987 dvtanlem 34002 dvtan 34003 ftc1cnnc 34027 dvasin 34039 dvacos 34040 dvreasin 34041 dvreacos 34042 areacirclem1 34043 areacirclem2 34044 areacirclem4 34046 refsumcn 40007 fprodcnlem 40626 fprodcn 40627 fsumcncf 40886 ioccncflimc 40893 cncfuni 40894 icocncflimc 40897 cncfdmsn 40898 cncfiooicclem1 40901 cxpcncf2 40908 fprodsub2cncf 40914 fprodadd2cncf 40915 dvmptconst 40924 dvmptidg 40926 dvresntr 40927 itgsubsticclem 40985 dirkercncflem2 41115 dirkercncflem4 41117 dirkercncf 41118 fourierdlem32 41150 fourierdlem33 41151 fourierdlem62 41179 fourierdlem93 41210 fourierdlem101 41218 |
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