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Mirrors > Home > ILE Home > Th. List > cnopncntop | GIF version |
Description: The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.) |
Ref | Expression |
---|---|
cnopncntop | ⊢ ℂ ∈ (MetOpen‘(abs ∘ − )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unicntopcntop 12923 | . 2 ⊢ ℂ = ∪ (MetOpen‘(abs ∘ − )) | |
2 | eqid 2157 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
3 | 2 | cntoptop 12920 | . . 3 ⊢ (MetOpen‘(abs ∘ − )) ∈ Top |
4 | ssid 3148 | . . 3 ⊢ (MetOpen‘(abs ∘ − )) ⊆ (MetOpen‘(abs ∘ − )) | |
5 | uniopn 12386 | . . 3 ⊢ (((MetOpen‘(abs ∘ − )) ∈ Top ∧ (MetOpen‘(abs ∘ − )) ⊆ (MetOpen‘(abs ∘ − ))) → ∪ (MetOpen‘(abs ∘ − )) ∈ (MetOpen‘(abs ∘ − ))) | |
6 | 3, 4, 5 | mp2an 423 | . 2 ⊢ ∪ (MetOpen‘(abs ∘ − )) ∈ (MetOpen‘(abs ∘ − )) |
7 | 1, 6 | eqeltri 2230 | 1 ⊢ ℂ ∈ (MetOpen‘(abs ∘ − )) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ⊆ wss 3102 ∪ cuni 3772 ∘ ccom 4589 ‘cfv 5169 ℂcc 7725 − cmin 8041 abscabs 10892 MetOpencmopn 12372 Topctop 12382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844 ax-pre-mulext 7845 ax-arch 7846 ax-caucvg 7847 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-map 6592 df-sup 6925 df-inf 6926 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-div 8541 df-inn 8829 df-2 8887 df-3 8888 df-4 8889 df-n0 9086 df-z 9163 df-uz 9435 df-q 9524 df-rp 9556 df-xneg 9674 df-xadd 9675 df-seqfrec 10340 df-exp 10414 df-cj 10737 df-re 10738 df-im 10739 df-rsqrt 10893 df-abs 10894 df-topgen 12359 df-psmet 12374 df-xmet 12375 df-met 12376 df-bl 12377 df-mopn 12378 df-top 12383 df-topon 12396 df-bases 12428 |
This theorem is referenced by: (None) |
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