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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 24803 | . 2 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
| 3 | 2 | topontopi 22921 | 1 ⊢ 𝐽 ∈ Top |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℂcc 11153 TopOpenctopn 17466 ℂfldccnfld 21364 Topctop 22899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 |
| This theorem is referenced by: cnopn 24807 rerest 24825 recld2 24836 zdis 24838 reperflem 24840 metdcn 24862 ngnmcncn 24867 metdscn2 24879 cncfcnvcn 24952 icchmeo 24971 icchmeoOLD 24972 cnrehmeo 24984 cnrehmeoOLD 24985 cnheiborlem 24986 cnheibor 24987 cnllycmp 24988 evth 24991 reparphti 25029 reparphtiOLD 25030 cncmet 25356 resscdrg 25392 mbfimaopn2 25692 ellimc2 25912 limcnlp 25913 limcflflem 25915 limcflf 25916 limccnp 25926 limciun 25929 dvbss 25936 perfdvf 25938 dvreslem 25944 dvres2lem 25945 dvidlem 25950 dvcnp2 25955 dvcnp2OLD 25956 dvnres 25967 dvaddbr 25974 dvmulbr 25975 dvmulbrOLD 25976 dvrec 25993 dvmptres 26001 dveflem 26017 dvlipcn 26033 dvcnvrelem2 26057 dvply1 26325 ulmdvlem3 26445 psercn 26470 abelth 26485 dvlog 26693 dvlog2 26695 efopnlem2 26699 efopn 26700 efrlim 27012 efrlimOLD 27013 lgamucov 27081 lgamucov2 27082 nmcnc 30715 raddcn 33928 lmlim 33946 cvxpconn 35247 cvxsconn 35248 cnllysconn 35250 ivthALT 36336 knoppcnlem10 36503 broucube 37661 binomcxplemdvbinom 44372 binomcxplemnotnn0 44375 climreeq 45628 limcrecl 45644 islpcn 45654 limcresiooub 45657 limcresioolb 45658 lptioo2cn 45660 lptioo1cn 45661 limclner 45666 fsumcncf 45893 ioccncflimc 45900 cncfuni 45901 icocncflimc 45904 cncfiooicclem1 45908 cncfiooicc 45909 itgsubsticclem 45990 dirkercncflem2 46119 dirkercncflem4 46121 dirkercncf 46122 fourierdlem32 46154 fourierdlem33 46155 fourierdlem48 46169 fourierdlem49 46170 fourierdlem62 46183 fourierdlem93 46214 fourierdlem101 46222 fourierdlem113 46234 fouriercnp 46241 fouriersw 46246 |
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