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Mirrors > Home > MPE Home > Th. List > unicntop | Structured version Visualization version GIF version |
Description: The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
unicntop | ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2797 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtopon 23078 | . 2 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
3 | 2 | toponunii 21212 | 1 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1525 ∪ cuni 4751 ‘cfv 6232 ℂcc 10388 TopOpenctopn 16528 ℂfldccnfld 20231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-fz 12747 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-plusg 16411 df-mulr 16412 df-starv 16413 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-rest 16529 df-topn 16530 df-topgen 16550 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-cnfld 20232 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-xms 22617 df-ms 22618 |
This theorem is referenced by: cnopn 23082 csscld 23539 clsocv 23540 cncmet 23612 resscdrg 23648 limciun 24179 dvidlem 24200 dvnres 24215 dvcjbr 24233 dvrec 24239 dvexp3 24262 dveflem 24263 lhop1lem 24297 dvply1 24560 psercn 24701 abelth 24716 logdmopn 24917 efrlim 25233 lgamucov2 25302 limcrecl 41473 islpcn 41483 lptioo2cn 41489 lptioo1cn 41490 limclner 41495 fsumcncf 41724 ioccncflimc 41731 cncfuni 41732 icocncflimc 41735 cncfiooicclem1 41739 itgsubsticclem 41823 dirkercncflem2 41953 dirkercncflem4 41955 fourierdlem32 41988 fourierdlem33 41989 fourierdlem62 42017 fourierdlem93 42048 fourierdlem101 42056 fourierdlem113 42068 fouriercnp 42075 |
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