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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 22391 | . 2 ⊢ ℂfld ∈ TopSp | |
2 | cnfldbas 19571 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
3 | cnfldtopn.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
4 | 2, 3 | istps 20551 | . 2 ⊢ (ℂfld ∈ TopSp ↔ 𝐽 ∈ (TopOn‘ℂ)) |
5 | 1, 4 | mpbi 219 | 1 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ‘cfv 5804 ℂcc 9813 TopOpenctopn 15905 ℂfldccnfld 19567 TopOnctopon 20518 TopSpctps 20519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-xms 21935 df-ms 21936 |
This theorem is referenced by: cnfldtop 22397 sszcld 22428 reperflem 22429 cnperf 22431 divcn 22479 fsumcn 22481 expcn 22483 divccn 22484 cncfcn1 22521 cncfmptc 22522 cncfmptid 22523 cncfmpt2f 22525 cdivcncf 22528 abscncfALT 22531 cncfcnvcn 22532 cnmptre 22534 iirevcn 22537 iihalf1cn 22539 iihalf2cn 22541 iimulcn 22545 icchmeo 22548 cnrehmeo 22560 cnheiborlem 22561 cnheibor 22562 cnllycmp 22563 evth 22566 evth2 22567 lebnumlem2 22569 reparphti 22605 pcoass 22632 csscld 22856 clsocv 22857 cncmet 22927 resscdrg 22962 mbfimaopnlem 23228 limcvallem 23441 ellimc2 23447 limcnlp 23448 limcflflem 23450 limcflf 23451 limcmo 23452 limcres 23456 cnplimc 23457 cnlimc 23458 limccnp 23461 limccnp2 23462 limciun 23464 dvbss 23471 perfdvf 23473 recnperf 23475 dvreslem 23479 dvres2lem 23480 dvres3a 23484 dvidlem 23485 dvcnp2 23489 dvcn 23490 dvnres 23500 dvaddbr 23507 dvmulbr 23508 dvcmulf 23514 dvcobr 23515 dvcjbr 23518 dvrec 23524 dvmptid 23526 dvmptc 23527 dvmptres2 23531 dvmptcmul 23533 dvmptntr 23540 dvmptfsum 23542 dvcnvlem 23543 dvcnv 23544 dvexp3 23545 dveflem 23546 dvlipcn 23561 lhop1lem 23580 lhop2 23582 lhop 23583 dvcnvrelem2 23585 dvcnvre 23586 ftc1lem3 23605 ftc1cn 23610 plycn 23821 dvply1 23843 dvtaylp 23928 taylthlem1 23931 taylthlem2 23932 ulmdvlem3 23960 psercn2 23981 psercn 23984 pserdvlem2 23986 pserdv 23987 abelth 23999 pige3 24073 logcn 24193 dvloglem 24194 logdmopn 24195 dvlog 24197 dvlog2 24199 efopnlem2 24203 efopn 24204 logtayl 24206 dvcxp1 24281 cxpcn 24286 cxpcn2 24287 cxpcn3 24289 resqrtcn 24290 sqrtcn 24291 loglesqrt 24299 atansopn 24459 dvatan 24462 xrlimcnp 24495 efrlim 24496 lgamucov 24564 lgamucov2 24565 ftalem3 24601 vmcn 26938 dipcn 26959 ipasslem7 27075 ipasslem8 27076 occllem 27546 nlelchi 28304 tpr2rico 29286 rmulccn 29302 raddcn 29303 cvxpcon 30478 cvxscon 30479 cnllyscon 30481 sinccvglem 30820 ivthALT 31500 knoppcnlem10 31662 knoppcnlem11 31663 broucube 32613 dvtanlem 32629 dvtan 32630 ftc1cnnc 32654 dvasin 32666 dvacos 32667 dvreasin 32668 dvreacos 32669 areacirclem1 32670 areacirclem2 32671 areacirclem4 32673 refsumcn 38212 unicntop 38230 fprodcnlem 38666 fprodcn 38667 fsumcncf 38763 ioccncflimc 38771 cncfuni 38772 icocncflimc 38775 cncfdmsn 38776 cncfiooicclem1 38779 cxpcncf2 38786 fprodsub2cncf 38792 fprodadd2cncf 38793 dvmptconst 38803 dvmptidg 38805 dvresntr 38806 itgsubsticclem 38867 dirkercncflem2 38997 dirkercncflem4 38999 dirkercncf 39000 fourierdlem32 39032 fourierdlem33 39033 fourierdlem62 39061 fourierdlem93 39092 fourierdlem101 39100 |
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