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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 22984 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
3 | eqid 2778 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
4 | 3 | mopntopon 22652 | . . 3 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)) |
5 | cnfldbas 20146 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
6 | cnfldtset 20150 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld) | |
7 | 5, 6 | topontopn 21152 | . . 3 ⊢ ((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) → (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld)) |
8 | 2, 4, 7 | mp2b 10 | . 2 ⊢ (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld) |
9 | 1, 8 | eqtr4i 2805 | 1 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ∘ ccom 5359 ‘cfv 6135 ℂcc 10270 − cmin 10606 abscabs 14381 TopOpenctopn 16468 ∞Metcxmet 20127 MetOpencmopn 20132 ℂfldccnfld 20142 TopOnctopon 21122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-bases 21158 |
This theorem is referenced by: cnfldhaus 22996 tgioo2 23014 recld2 23025 zdis 23027 reperflem 23029 addcnlem 23075 divcn 23079 dfii3 23094 cncfcn 23120 cnheibor 23162 cnllycmp 23163 ipcn 23452 lmclim 23509 cncmet 23528 recmet 23529 ellimc3 24080 dvlipcn 24194 lhop1lem 24213 ftc1lem6 24241 ulmdvlem3 24593 psercn 24617 pserdvlem2 24619 abelth 24632 dvlog2 24836 efopnlem2 24840 efopn 24841 logtayl 24843 cxpcn3 24929 rlimcnp 25144 xrlimcnp 25147 efrlim 25148 lgamucov 25216 ftalem3 25253 smcnlem 28124 hhcnf 29336 tpr2rico 30556 cnllysconn 31826 ftc1cnnc 34109 binomcxplemdvbinom 39508 binomcxplemnotnn0 39511 limcrecl 40769 islpcn 40779 |
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