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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 24687 | . 2 ⊢ ℂfld ∈ TopSp | |
| 2 | cnfldbas 21276 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
| 3 | cnfldtopn.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
| 4 | 2, 3 | istps 22829 | . 2 ⊢ (ℂfld ∈ TopSp ↔ 𝐽 ∈ (TopOn‘ℂ)) |
| 5 | 1, 4 | mpbi 229 | 1 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6542 ℂcc 11130 TopOpenctopn 17396 ℂfldccnfld 21272 TopOnctopon 22805 TopSpctps 22827 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-fz 13511 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-starv 17241 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-rest 17397 df-topn 17398 df-topgen 17418 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-xms 24219 df-ms 24220 |
| This theorem is referenced by: cnfldtop 24693 unicntop 24695 sszcld 24726 reperflem 24727 cnperf 24729 divcnOLD 24777 divcn 24779 fsumcn 24781 expcn 24783 divccn 24784 expcnOLD 24785 divccnOLD 24786 cncfcn1 24824 cncfmptc 24825 cncfmptid 24826 cncfmpt2f 24828 cdivcncf 24834 abscncfALT 24838 cncfcnvcn 24839 cnmptre 24841 iirevcn 24844 iihalf1cn 24846 iihalf1cnOLD 24847 iihalf2cn 24849 iihalf2cnOLD 24850 iimulcn 24854 iimulcnOLD 24855 icchmeo 24858 icchmeoOLD 24859 cnrehmeo 24871 cnrehmeoOLD 24872 cnheiborlem 24873 cnheibor 24874 cnllycmp 24875 evth 24878 evth2 24879 lebnumlem2 24881 reparphti 24916 reparphtiOLD 24917 pcoass 24944 mulcncf 25367 mbfimaopnlem 25577 limcvallem 25793 ellimc2 25799 limcnlp 25800 limcflflem 25802 limcflf 25803 limcmo 25804 limcres 25808 cnplimc 25809 cnlimc 25810 limccnp 25813 limccnp2 25814 dvbss 25823 perfdvf 25825 recnperf 25827 dvreslem 25831 dvres2lem 25832 dvres3a 25836 dvidlem 25837 dvcnp2 25842 dvcnp2OLD 25843 dvcn 25844 dvnres 25854 dvaddbr 25861 dvmulbr 25862 dvmulbrOLD 25863 dvcmulf 25869 dvcobr 25870 dvcobrOLD 25871 dvcjbr 25874 dvrec 25880 dvmptid 25882 dvmptc 25883 dvmptres2 25887 dvmptcmul 25889 dvmptntr 25896 dvmptfsum 25900 dvcnvlem 25901 dvcnv 25902 dvexp3 25903 dveflem 25904 dvlipcn 25920 lhop1lem 25939 lhop2 25941 lhop 25942 dvcnvrelem2 25944 dvcnvre 25945 ftc1lem3 25966 ftc1cn 25971 plycn 26188 plycnOLD 26189 dvply1 26211 dvtaylp 26298 taylthlem1 26301 taylthlem2 26302 taylthlem2OLD 26303 ulmdvlem3 26331 psercn2 26352 psercn2OLD 26353 psercn 26356 pserdvlem2 26358 pserdv 26359 abelth 26371 pige3ALT 26447 logcn 26574 dvloglem 26575 dvlog 26578 dvlog2 26580 efopnlem2 26584 efopn 26585 logtayl 26587 dvcxp1 26667 cxpcn 26672 cxpcnOLD 26673 cxpcn2 26674 cxpcn3 26676 resqrtcn 26677 sqrtcn 26678 loglesqrt 26686 atansopn 26857 dvatan 26860 xrlimcnp 26893 efrlim 26894 efrlimOLD 26895 lgamucov 26963 ftalem3 27000 vmcn 30502 dipcn 30523 ipasslem7 30639 ipasslem8 30640 occllem 31106 nlelchi 31864 tpr2rico 33507 rmulccn 33523 raddcn 33524 cxpcncf1 34221 cvxpconn 34846 cvxsconn 34847 cnllysconn 34849 sinccvglem 35270 ivthALT 35813 knoppcnlem10 35971 knoppcnlem11 35972 broucube 37121 dvtanlem 37136 dvtan 37137 ftc1cnnc 37159 dvasin 37171 dvacos 37172 dvreasin 37173 dvreacos 37174 areacirclem1 37175 areacirclem2 37176 areacirclem4 37178 refsumcn 44386 fprodcnlem 44981 fprodcn 44982 fsumcncf 45260 ioccncflimc 45267 cncfuni 45268 icocncflimc 45271 cncfdmsn 45272 cncfiooicclem1 45275 cxpcncf2 45281 fprodsub2cncf 45287 fprodadd2cncf 45288 dvmptconst 45297 dvmptidg 45299 dvresntr 45300 itgsubsticclem 45357 dirkercncflem2 45486 dirkercncflem4 45488 dirkercncf 45489 fourierdlem32 45521 fourierdlem33 45522 fourierdlem62 45550 fourierdlem93 45581 fourierdlem101 45589 |
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