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Mirrors > Home > MPE Home > Th. List > cnfldtop | 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 |
---|---|
cnfldtop | ⊢ 𝐽 ∈ Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldtopn.1 | . . 3 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtopon 23955 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
3 | 2 | topontopi 22073 | 1 ⊢ 𝐽 ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2107 ‘cfv 6437 ℂcc 10878 TopOpenctopn 17141 ℂfldccnfld 20606 Topctop 22051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-er 8507 df-map 8626 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-sup 9210 df-inf 9211 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-z 12329 df-dec 12447 df-uz 12592 df-q 12698 df-rp 12740 df-xneg 12857 df-xadd 12858 df-xmul 12859 df-fz 13249 df-seq 13731 df-exp 13792 df-cj 14819 df-re 14820 df-im 14821 df-sqrt 14955 df-abs 14956 df-struct 16857 df-slot 16892 df-ndx 16904 df-base 16922 df-plusg 16984 df-mulr 16985 df-starv 16986 df-tset 16990 df-ple 16991 df-ds 16993 df-unif 16994 df-rest 17142 df-topn 17143 df-topgen 17163 df-psmet 20598 df-xmet 20599 df-met 20600 df-bl 20601 df-mopn 20602 df-cnfld 20607 df-top 22052 df-topon 22069 df-topsp 22091 df-bases 22105 df-xms 23482 df-ms 23483 |
This theorem is referenced by: cnopn 23959 rerest 23976 recld2 23986 zdis 23988 reperflem 23990 metdcn 24012 ngnmcncn 24017 metdscn2 24029 cncfcnvcn 24097 icchmeo 24113 cnrehmeo 24125 cnheiborlem 24126 cnheibor 24127 cnllycmp 24128 evth 24131 reparphti 24169 cncmet 24495 resscdrg 24531 mbfimaopn2 24830 ellimc2 25050 limcnlp 25051 limcflflem 25053 limcflf 25054 limccnp 25064 limciun 25067 dvbss 25074 perfdvf 25076 dvreslem 25082 dvres2lem 25083 dvidlem 25088 dvcnp2 25093 dvnres 25104 dvaddbr 25111 dvmulbr 25112 dvrec 25128 dvmptres 25136 dveflem 25152 dvlipcn 25167 dvcnvrelem2 25191 dvply1 25453 ulmdvlem3 25570 psercn 25594 abelth 25609 dvlog 25815 dvlog2 25817 efopnlem2 25821 efopn 25822 efrlim 26128 lgamucov 26196 lgamucov2 26197 nmcnc 29067 raddcn 31888 lmlim 31906 cvxpconn 33213 cvxsconn 33214 cnllysconn 33216 ivthALT 34533 broucube 35820 binomcxplemdvbinom 41978 binomcxplemnotnn0 41981 climreeq 43161 limcrecl 43177 islpcn 43187 limcresiooub 43190 limcresioolb 43191 lptioo2cn 43193 lptioo1cn 43194 limclner 43199 fsumcncf 43426 ioccncflimc 43433 cncfuni 43434 icocncflimc 43437 cncfiooicclem1 43441 cncfiooicc 43442 itgsubsticclem 43523 dirkercncflem2 43652 dirkercncflem4 43654 dirkercncf 43655 fourierdlem32 43687 fourierdlem33 43688 fourierdlem48 43702 fourierdlem49 43703 fourierdlem62 43716 fourierdlem93 43747 fourierdlem101 43755 fourierdlem113 43767 fouriercnp 43774 fouriersw 43779 |
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