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Theorem cnfldtopn 24742

Description: The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)

**Hypothesis**
Ref	Expression
cnfldtopn.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
cnfldtopn	⊢ 𝐽 = (MetOpen‘(abs ∘ − ))

**Proof of Theorem cnfldtopn**
Step	Hyp	Ref	Expression
1		cnfldtopn.1	. 2 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
2		cnxmet 24733	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		eqid 2725	. . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − ))
4	3	mopntopon 24389	. . 3 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ))
5		cnfldbas 21300	. . . 4 ⊢ ℂ = (Base‘ℂ_fld)
6		cnfldtset 21306	. . . 4 ⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂ_fld)
7	5, 6	topontopn 22886	. . 3 ⊢ ((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) → (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂ_fld))
8	2, 4, 7	mp2b 10	. 2 ⊢ (MetOpen‘(abs ∘ − )) = (TopOpen‘ℂ_fld)
9	1, 8	eqtr4i 2756	1 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ∘ ccom 5682 ‘cfv 6549 ℂcc 11138 − cmin 11476 abscabs 15217 TopOpenctopn 17406 ∞Metcxmet 21281 MetOpencmopn 21286 ℂ_fldccnfld 21296 TopOnctopon 22856

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-bases 22893

This theorem is referenced by: cnfldhaus 24745 tgioo2 24763 recld2 24774 zdis 24776 reperflem 24778 addcnlem 24824 divcnOLD 24828 divcn 24830 dfii3 24847 cncfcn 24874 cnheibor 24925 cnllycmp 24926 ipcn 25218 lmclim 25275 cncmet 25294 recmet 25295 ellimc3 25852 dvlipcn 25971 lhop1lem 25990 ftc1lem6 26020 ulmdvlem3 26383 psercn 26408 pserdvlem2 26410 abelth 26423 dvlog2 26632 efopnlem2 26636 efopn 26637 logtayl 26639 cxpcn3 26728 rlimcnp 26942 xrlimcnp 26945 efrlim 26946 efrlimOLD 26947 lgamucov 27015 ftalem3 27052 smcnlem 30579 hhcnf 31787 tpr2rico 33644 cnllysconn 34986 ftc1cnnc 37296 binomcxplemdvbinom 43932 binomcxplemnotnn0 43935 limcrecl 45155 islpcn 45165

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