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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 23388 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
3 | 2 | topontopi 21520 | 1 ⊢ 𝐽 ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ‘cfv 6324 ℂcc 10524 TopOpenctopn 16687 ℂfldccnfld 20091 Topctop 21498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-xms 22927 df-ms 22928 |
This theorem is referenced by: cnopn 23392 rerest 23409 recld2 23419 zdis 23421 reperflem 23423 metdcn 23445 ngnmcncn 23450 metdscn2 23462 cncfcnvcn 23530 icchmeo 23546 cnrehmeo 23558 cnheiborlem 23559 cnheibor 23560 cnllycmp 23561 evth 23564 reparphti 23602 cncmet 23926 resscdrg 23962 mbfimaopn2 24261 ellimc2 24480 limcnlp 24481 limcflflem 24483 limcflf 24484 limccnp 24494 limciun 24497 dvbss 24504 perfdvf 24506 dvreslem 24512 dvres2lem 24513 dvidlem 24518 dvcnp2 24523 dvnres 24534 dvaddbr 24541 dvmulbr 24542 dvrec 24558 dvmptres 24566 dveflem 24582 dvlipcn 24597 dvcnvrelem2 24621 dvply1 24880 ulmdvlem3 24997 psercn 25021 abelth 25036 dvlog 25242 dvlog2 25244 efopnlem2 25248 efopn 25249 efrlim 25555 lgamucov 25623 lgamucov2 25624 nmcnc 28479 raddcn 31282 lmlim 31300 cvxpconn 32602 cvxsconn 32603 cnllysconn 32605 ivthALT 33796 broucube 35091 binomcxplemdvbinom 41057 binomcxplemnotnn0 41060 climreeq 42255 limcrecl 42271 islpcn 42281 limcresiooub 42284 limcresioolb 42285 lptioo2cn 42287 lptioo1cn 42288 limclner 42293 fsumcncf 42520 ioccncflimc 42527 cncfuni 42528 icocncflimc 42531 cncfiooicclem1 42535 cncfiooicc 42536 itgsubsticclem 42617 dirkercncflem2 42746 dirkercncflem4 42748 dirkercncf 42749 fourierdlem32 42781 fourierdlem33 42782 fourierdlem48 42796 fourierdlem49 42797 fourierdlem62 42810 fourierdlem93 42841 fourierdlem101 42849 fourierdlem113 42861 fouriercnp 42868 fouriersw 42873 |
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