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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 24707 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
3 | eqid 2725 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
4 | 3 | mopntopon 24363 | . . 3 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ)) |
5 | cnfldbas 21287 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
6 | cnfldtset 21293 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld) | |
7 | 5, 6 | topontopn 22860 | . . 3 ⊢ ((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) → (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld)) |
8 | 2, 4, 7 | mp2b 10 | . 2 ⊢ (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂfld) |
9 | 1, 8 | eqtr4i 2756 | 1 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∘ ccom 5676 ‘cfv 6543 ℂcc 11136 − cmin 11474 abscabs 15213 TopOpenctopn 17402 ∞Metcxmet 21268 MetOpencmopn 21273 ℂfldccnfld 21283 TopOnctopon 22830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-topn 17404 df-topgen 17424 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22814 df-topon 22831 df-bases 22867 |
This theorem is referenced by: cnfldhaus 24719 tgioo2 24737 recld2 24748 zdis 24750 reperflem 24752 addcnlem 24798 divcnOLD 24802 divcn 24804 dfii3 24821 cncfcn 24848 cnheibor 24899 cnllycmp 24900 ipcn 25192 lmclim 25249 cncmet 25268 recmet 25269 ellimc3 25826 dvlipcn 25945 lhop1lem 25964 ftc1lem6 25994 ulmdvlem3 26356 psercn 26381 pserdvlem2 26383 abelth 26396 dvlog2 26605 efopnlem2 26609 efopn 26610 logtayl 26612 cxpcn3 26701 rlimcnp 26915 xrlimcnp 26918 efrlim 26919 efrlimOLD 26920 lgamucov 26988 ftalem3 27025 smcnlem 30551 hhcnf 31759 tpr2rico 33570 cnllysconn 34912 ftc1cnnc 37222 binomcxplemdvbinom 43855 binomcxplemnotnn0 43858 limcrecl 45080 islpcn 45090 |
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