tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 23700

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 2737	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 23670	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 10786	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 23679	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2737	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 23422	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 692	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 23693	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 × cxp 5549 ran crn 5552 ↾ cres 5553 ∘ ccom 5555 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 − cmin 11062 (,)cioo 12935 abscabs 14797 ↾_t crest 16925 TopOpenctopn 16926 topGenctg 16942 ∞Metcxmet 20348 MetOpencmopn 20353 ℂ_fldccnfld 20363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-fz 13096 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-rest 16927 df-topn 16928 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-bases 21843

This theorem is referenced by: rerest 23701 tgioo3 23702 zcld2 23712 metdcn 23737 ngnmcncn 23742 metdscn2 23754 abscncfALT 23821 cnrehmeo 23850 rellycmp 23854 evth 23856 evth2 23857 lebnumlem2 23859 resscdrg 24255 retopn 24276 cncombf 24555 cnmbf 24556 dvmptresicc 24813 dvcjbr 24846 rolle 24887 cmvth 24888 mvth 24889 dvlip 24890 dvlipcn 24891 dvlip2 24892 c1liplem1 24893 dvgt0lem1 24899 dvle 24904 dvivthlem1 24905 dvne0 24908 lhop1lem 24910 lhop2 24912 lhop 24913 dvcnvrelem1 24914 dvcnvrelem2 24915 dvcnvre 24916 dvcvx 24917 dvfsumle 24918 dvfsumabs 24920 dvfsumlem2 24924 ftc1 24939 ftc1cn 24940 ftc2 24941 ftc2ditglem 24942 itgparts 24944 itgsubstlem 24945 itgpowd 24947 taylthlem2 25266 efcvx 25341 pige3ALT 25409 dvloglem 25536 logdmopn 25537 advlog 25542 advlogexp 25543 logccv 25551 loglesqrt 25644 lgamgulmlem2 25912 ftalem3 25957 log2sumbnd 26425 nmcnc 28777 ipasslem7 28917 rmulccn 31592 raddcn 31593 ftc2re 32290 knoppcnlem10 34419 knoppcnlem11 34420 broucube 35548 ftc1cnnc 35586 ftc2nc 35596 dvasin 35598 dvacos 35599 dvreasin 35600 dvreacos 35601 areacirclem1 35602 areacirc 35607 dvrelog2 39805 dvrelog3 39806 aks4d1p1p6 39814 lhe4.4ex1a 41620 refsumcn 42246 xrtgcntopre 42694 tgioo4 42786 climreeq 42829 limcresiooub 42858 limcresioolb 42859 lptioo2cn 42861 lptioo1cn 42862 limclner 42867 cncfiooicclem1 43109 jumpncnp 43114 dvresioo 43137 dvbdfbdioolem1 43144 itgsin0pilem1 43166 itgsinexplem1 43170 itgsubsticclem 43191 itgiccshift 43196 itgperiod 43197 itgsbtaddcnst 43198 dirkeritg 43318 dirkercncflem2 43320 dirkercncflem3 43321 dirkercncflem4 43322 dirkercncf 43323 fourierdlem28 43351 fourierdlem32 43355 fourierdlem33 43356 fourierdlem39 43362 fourierdlem56 43378 fourierdlem57 43379 fourierdlem58 43380 fourierdlem59 43381 fourierdlem62 43384 fourierdlem68 43390 fourierdlem72 43394 fourierdlem73 43395 fourierdlem74 43396 fourierdlem75 43397 fourierdlem80 43402 fourierdlem94 43416 fourierdlem103 43425 fourierdlem104 43426 fourierdlem113 43435 fouriercnp 43442 fouriersw 43447 fouriercn 43448 etransclem2 43452 etransclem23 43473 etransclem35 43485 etransclem38 43488 etransclem39 43489 etransclem44 43494 etransclem45 43495 etransclem46 43496 etransclem47 43497

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