![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22633 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
3 | 2 | topontopi 20768 | 1 ⊢ 𝐽 ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 ‘cfv 5926 ℂcc 9972 TopOpenctopn 16129 ℂfldccnfld 19794 Topctop 20746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-fz 12365 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-plusg 16001 df-mulr 16002 df-starv 16003 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-rest 16130 df-topn 16131 df-topgen 16151 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-xms 22172 df-ms 22173 |
This theorem is referenced by: cnopn 22637 rerest 22654 recld2 22664 zdis 22666 reperflem 22668 metdcn 22690 ngnmcncn 22695 metdscn2 22707 cncfcn1 22760 cncfcnvcn 22771 icchmeo 22787 cnrehmeo 22799 cnheiborlem 22800 cnheibor 22801 cnllycmp 22802 evth 22805 reparphti 22843 cncmet 23165 resscdrg 23200 mbfimaopn2 23469 ellimc2 23686 limcnlp 23687 limcflflem 23689 limcflf 23690 limccnp 23700 limciun 23703 dvbss 23710 perfdvf 23712 dvreslem 23718 dvres2lem 23719 dvidlem 23724 dvcnp2 23728 dvnres 23739 dvaddbr 23746 dvmulbr 23747 dvrec 23763 dvmptres 23771 dvexp3 23786 dveflem 23787 dvlipcn 23802 dvcnvrelem2 23826 ftc1cn 23851 dvply1 24084 ulmdvlem3 24201 psercn 24225 pserdvlem2 24227 pserdv 24228 abelth 24240 logcn 24438 dvloglem 24439 dvlog 24442 dvlog2 24444 efopnlem2 24448 efopn 24449 logtayl 24451 dvatan 24707 efrlim 24741 lgamucov 24809 lgamucov2 24810 ftalem3 24846 nmcnc 27679 raddcn 30103 lmlim 30121 cvxpconn 31350 cvxsconn 31351 cnllysconn 31353 ivthALT 32455 broucube 33573 ftc1cnnc 33614 binomcxplemdvbinom 38869 binomcxplemnotnn0 38872 climreeq 40163 limcrecl 40179 islpcn 40189 limcresiooub 40192 limcresioolb 40193 lptioo2cn 40195 lptioo1cn 40196 limclner 40201 fsumcncf 40409 ioccncflimc 40416 cncfuni 40417 icocncflimc 40420 cncfiooicclem1 40424 cncfiooicc 40425 itgsubsticclem 40509 dirkercncflem2 40639 dirkercncflem4 40641 dirkercncf 40642 fourierdlem32 40674 fourierdlem33 40675 fourierdlem48 40689 fourierdlem49 40690 fourierdlem62 40703 fourierdlem93 40734 fourierdlem101 40742 fourierdlem113 40754 fouriercnp 40761 fouriersw 40766 |
Copyright terms: Public domain | W3C validator |