tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 23872

Description: The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)

**Hypothesis**
Ref	Expression
tgioo2.1	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
tgioo2	⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

**Proof of Theorem tgioo2**
Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 23842	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 10859	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 23851	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2738	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 23586	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 688	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 23865	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 × cxp 5578 ran crn 5581 ↾ cres 5582 ∘ ccom 5584 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 − cmin 11135 (,)cioo 13008 abscabs 14873 ↾_t crest 17048 TopOpenctopn 17049 topGenctg 17065 ∞Metcxmet 20495 MetOpencmopn 20500 ℂ_fldccnfld 20510

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-bases 22004

This theorem is referenced by: rerest 23873 tgioo3 23874 zcld2 23884 metdcn 23909 ngnmcncn 23914 metdscn2 23926 abscncfALT 23993 cnrehmeo 24022 rellycmp 24026 evth 24028 evth2 24029 lebnumlem2 24031 resscdrg 24427 retopn 24448 cncombf 24727 cnmbf 24728 dvmptresicc 24985 dvcjbr 25018 rolle 25059 cmvth 25060 mvth 25061 dvlip 25062 dvlipcn 25063 dvlip2 25064 c1liplem1 25065 dvgt0lem1 25071 dvle 25076 dvivthlem1 25077 dvne0 25080 lhop1lem 25082 lhop2 25084 lhop 25085 dvcnvrelem1 25086 dvcnvrelem2 25087 dvcnvre 25088 dvcvx 25089 dvfsumle 25090 dvfsumabs 25092 dvfsumlem2 25096 ftc1 25111 ftc1cn 25112 ftc2 25113 ftc2ditglem 25114 itgparts 25116 itgsubstlem 25117 itgpowd 25119 taylthlem2 25438 efcvx 25513 pige3ALT 25581 dvloglem 25708 logdmopn 25709 advlog 25714 advlogexp 25715 logccv 25723 loglesqrt 25816 lgamgulmlem2 26084 ftalem3 26129 log2sumbnd 26597 nmcnc 28959 ipasslem7 29099 rmulccn 31780 raddcn 31781 ftc2re 32478 knoppcnlem10 34609 knoppcnlem11 34610 broucube 35738 ftc1cnnc 35776 ftc2nc 35786 dvasin 35788 dvacos 35789 dvreasin 35790 dvreacos 35791 areacirclem1 35792 areacirc 35797 dvrelog2 40000 dvrelog3 40001 aks4d1p1p6 40009 lhe4.4ex1a 41836 refsumcn 42462 xrtgcntopre 42909 tgioo4 43001 climreeq 43044 limcresiooub 43073 limcresioolb 43074 lptioo2cn 43076 lptioo1cn 43077 limclner 43082 cncfiooicclem1 43324 jumpncnp 43329 dvresioo 43352 dvbdfbdioolem1 43359 itgsin0pilem1 43381 itgsinexplem1 43385 itgsubsticclem 43406 itgiccshift 43411 itgperiod 43412 itgsbtaddcnst 43413 dirkeritg 43533 dirkercncflem2 43535 dirkercncflem3 43536 dirkercncflem4 43537 dirkercncf 43538 fourierdlem28 43566 fourierdlem32 43570 fourierdlem33 43571 fourierdlem39 43577 fourierdlem56 43593 fourierdlem57 43594 fourierdlem58 43595 fourierdlem59 43596 fourierdlem62 43599 fourierdlem68 43605 fourierdlem72 43609 fourierdlem73 43610 fourierdlem74 43611 fourierdlem75 43612 fourierdlem80 43617 fourierdlem94 43631 fourierdlem103 43640 fourierdlem104 43641 fourierdlem113 43650 fouriercnp 43657 fouriersw 43662 fouriercn 43663 etransclem2 43667 etransclem23 43688 etransclem35 43700 etransclem38 43703 etransclem39 43704 etransclem44 43709 etransclem45 43710 etransclem46 43711 etransclem47 43712

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