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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 23391 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
3 | 2 | topontopi 21523 | 1 ⊢ 𝐽 ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ‘cfv 6355 ℂcc 10535 TopOpenctopn 16695 ℂfldccnfld 20545 Topctop 21501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-xms 22930 df-ms 22931 |
This theorem is referenced by: cnopn 23395 rerest 23412 recld2 23422 zdis 23424 reperflem 23426 metdcn 23448 ngnmcncn 23453 metdscn2 23465 cncfcnvcn 23529 icchmeo 23545 cnrehmeo 23557 cnheiborlem 23558 cnheibor 23559 cnllycmp 23560 evth 23563 reparphti 23601 cncmet 23925 resscdrg 23961 mbfimaopn2 24258 ellimc2 24475 limcnlp 24476 limcflflem 24478 limcflf 24479 limccnp 24489 limciun 24492 dvbss 24499 perfdvf 24501 dvreslem 24507 dvres2lem 24508 dvidlem 24513 dvcnp2 24517 dvnres 24528 dvaddbr 24535 dvmulbr 24536 dvrec 24552 dvmptres 24560 dveflem 24576 dvlipcn 24591 dvcnvrelem2 24615 dvply1 24873 ulmdvlem3 24990 psercn 25014 abelth 25029 dvlog 25234 dvlog2 25236 efopnlem2 25240 efopn 25241 efrlim 25547 lgamucov 25615 lgamucov2 25616 nmcnc 28473 raddcn 31172 lmlim 31190 cvxpconn 32489 cvxsconn 32490 cnllysconn 32492 ivthALT 33683 broucube 34941 binomcxplemdvbinom 40705 binomcxplemnotnn0 40708 climreeq 41914 limcrecl 41930 islpcn 41940 limcresiooub 41943 limcresioolb 41944 lptioo2cn 41946 lptioo1cn 41947 limclner 41952 fsumcncf 42181 ioccncflimc 42188 cncfuni 42189 icocncflimc 42192 cncfiooicclem1 42196 cncfiooicc 42197 itgsubsticclem 42280 dirkercncflem2 42409 dirkercncflem4 42411 dirkercncf 42412 fourierdlem32 42444 fourierdlem33 42445 fourierdlem48 42459 fourierdlem49 42460 fourierdlem62 42473 fourierdlem93 42504 fourierdlem101 42512 fourierdlem113 42524 fouriercnp 42531 fouriersw 42536 |
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