tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 22606

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 2622	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 22576	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 9993	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 22585	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2622	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 22329	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 708	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 22599	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 × cxp 5112 ran crn 5115 ↾ cres 5116 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 − cmin 10266 (,)cioo 12175 abscabs 13974 ↾_t crest 16081 TopOpenctopn 16082 topGenctg 16098 ∞Metcxmt 19731 MetOpencmopn 19736 ℂ_fldccnfld 19746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-bases 20750

This theorem is referenced by: rerest 22607 tgioo3 22608 zcld2 22618 metdcn 22643 ngnmcncn 22648 metdscn2 22660 abscncfALT 22723 cnrehmeo 22752 rellycmp 22756 evth 22758 evth2 22759 lebnumlem2 22761 resscdrg 23154 retopn 23167 cncombf 23425 cnmbf 23426 dvcjbr 23712 rolle 23753 cmvth 23754 mvth 23755 dvlip 23756 dvlipcn 23757 dvlip2 23758 c1liplem1 23759 dvgt0lem1 23765 dvle 23770 dvivthlem1 23771 dvne0 23774 lhop1lem 23776 lhop2 23778 lhop 23779 dvcnvrelem1 23780 dvcnvrelem2 23781 dvcnvre 23782 dvcvx 23783 dvfsumle 23784 dvfsumabs 23786 dvfsumlem2 23790 ftc1 23805 ftc1cn 23806 ftc2 23807 ftc2ditglem 23808 itgparts 23810 itgsubstlem 23811 taylthlem2 24128 efcvx 24203 pige3 24269 dvloglem 24394 logdmopn 24395 advlog 24400 advlogexp 24401 logccv 24409 loglesqrt 24499 lgamgulmlem2 24756 ftalem3 24801 log2sumbnd 25233 nmcnc 27551 ipasslem7 27691 rmulccn 29974 raddcn 29975 ftc2re 30676 knoppcnlem10 32492 knoppcnlem11 32493 broucube 33443 ftc1cnnc 33484 ftc2nc 33494 dvasin 33496 dvacos 33497 dvreasin 33498 dvreacos 33499 areacirclem1 33500 areacirc 33505 itgpowd 37800 lhe4.4ex1a 38528 refsumcn 39189 xrtgcntopre 39709 tgioo4 39800 climreeq 39845 limcresiooub 39874 limcresioolb 39875 lptioo2cn 39877 lptioo1cn 39878 limclner 39883 cncfiooicclem1 40106 jumpncnp 40111 fperdvper 40133 dvmptresicc 40134 dvresioo 40136 dvbdfbdioolem1 40143 itgsin0pilem1 40165 itgsinexplem1 40169 itgcoscmulx 40185 itgsubsticclem 40191 itgiccshift 40196 itgperiod 40197 itgsbtaddcnst 40198 dirkeritg 40319 dirkercncflem2 40321 dirkercncflem3 40322 dirkercncflem4 40323 dirkercncf 40324 fourierdlem28 40352 fourierdlem32 40356 fourierdlem33 40357 fourierdlem39 40363 fourierdlem56 40379 fourierdlem57 40380 fourierdlem58 40381 fourierdlem59 40382 fourierdlem60 40383 fourierdlem61 40384 fourierdlem62 40385 fourierdlem68 40391 fourierdlem72 40395 fourierdlem73 40396 fourierdlem74 40397 fourierdlem75 40398 fourierdlem80 40403 fourierdlem94 40417 fourierdlem103 40426 fourierdlem104 40427 fourierdlem113 40436 fouriercnp 40443 fouriersw 40448 fouriercn 40449 etransclem2 40453 etransclem23 40474 etransclem35 40486 etransclem38 40489 etransclem39 40490 etransclem44 40495 etransclem45 40496 etransclem46 40497 etransclem47 40498

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