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Mirrors > Home > MPE Home > Th. List > cnfldtopn | Structured version Visualization version GIF version |
Description: The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnfldtopn.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
cnfldtopn | ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldtopn.1 | . 2 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
2 | cnxmet 23378 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
3 | eqid 2798 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
4 | 3 | mopntopon 23046 | . . 3 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)) |
5 | cnfldbas 20095 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
6 | cnfldtset 20099 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld) | |
7 | 5, 6 | topontopn 21545 | . . 3 ⊢ ((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) → (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld)) |
8 | 2, 4, 7 | mp2b 10 | . 2 ⊢ (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld) |
9 | 1, 8 | eqtr4i 2824 | 1 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∘ ccom 5523 ‘cfv 6324 ℂcc 10524 − cmin 10859 abscabs 14585 TopOpenctopn 16687 ∞Metcxmet 20076 MetOpencmopn 20081 ℂfldccnfld 20091 TopOnctopon 21515 |
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-bases 21551 |
This theorem is referenced by: cnfldhaus 23390 tgioo2 23408 recld2 23419 zdis 23421 reperflem 23423 addcnlem 23469 divcn 23473 dfii3 23488 cncfcn 23515 cnheibor 23560 cnllycmp 23561 ipcn 23850 lmclim 23907 cncmet 23926 recmet 23927 ellimc3 24482 dvlipcn 24597 lhop1lem 24616 ftc1lem6 24644 ulmdvlem3 24997 psercn 25021 pserdvlem2 25023 abelth 25036 dvlog2 25244 efopnlem2 25248 efopn 25249 logtayl 25251 cxpcn3 25337 rlimcnp 25551 xrlimcnp 25554 efrlim 25555 lgamucov 25623 ftalem3 25660 smcnlem 28480 hhcnf 29688 tpr2rico 31265 cnllysconn 32605 ftc1cnnc 35129 binomcxplemdvbinom 41057 binomcxplemnotnn0 41060 limcrecl 42271 islpcn 42281 |
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