tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 24807

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 2726	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 24777	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 11206	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 24786	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2726	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 24521	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 690	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 24800	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 × cxp 5672 ran crn 5675 ↾ cres 5676 ∘ ccom 5678 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 ℝcr 11148 − cmin 11485 (,)cioo 13372 abscabs 15234 ↾_t crest 17430 TopOpenctopn 17431 topGenctg 17447 ∞Metcxmet 21324 MetOpencmopn 21329 ℂ_fldccnfld 21339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-fz 13533 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-slot 17179 df-ndx 17191 df-base 17209 df-plusg 17274 df-mulr 17275 df-starv 17276 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-rest 17432 df-topn 17433 df-topgen 17453 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-cnfld 21340 df-top 22884 df-topon 22901 df-bases 22937

This theorem is referenced by: rerest 24808 tgioo3 24809 zcld2 24819 metdcn 24844 ngnmcncn 24849 metdscn2 24861 abscncfALT 24933 cnrehmeo 24966 cnrehmeoOLD 24967 rellycmp 24971 evth 24973 evth2 24974 lebnumlem2 24976 resscdrg 25374 retopn 25395 cncombf 25675 cnmbf 25676 dvmptresicc 25933 dvcjbr 25969 rolle 26010 cmvth 26011 cmvthOLD 26012 mvth 26013 dvlip 26014 dvlipcn 26015 dvlip2 26016 c1liplem1 26017 dvgt0lem1 26023 dvle 26028 dvivthlem1 26029 dvne0 26032 lhop1lem 26034 lhop2 26036 lhop 26037 dvcnvrelem1 26038 dvcnvrelem2 26039 dvcnvre 26040 dvcvx 26041 dvfsumle 26042 dvfsumleOLD 26043 dvfsumabs 26045 dvfsumlem2 26049 dvfsumlem2OLD 26050 ftc1 26065 ftc1cn 26066 ftc2 26067 ftc2ditglem 26068 itgparts 26070 itgsubstlem 26071 itgpowd 26073 taylthlem2 26399 taylthlem2OLD 26400 efcvx 26476 pige3ALT 26544 dvloglem 26672 logdmopn 26673 advlog 26678 advlogexp 26679 logccv 26687 loglesqrt 26786 lgamgulmlem2 27055 ftalem3 27100 log2sumbnd 27570 nmcnc 30626 ipasslem7 30766 rmulccn 33756 raddcn 33757 ftc2re 34457 knoppcnlem10 36218 knoppcnlem11 36219 broucube 37368 ftc1cnnc 37406 ftc2nc 37416 dvasin 37418 dvacos 37419 dvreasin 37420 dvreacos 37421 areacirclem1 37422 areacirc 37427 dvrelog2 41776 dvrelog3 41777 aks4d1p1p6 41785 lhe4.4ex1a 44040 refsumcn 44666 xrtgcntopre 45130 tgioo4 45227 climreeq 45270 limcresiooub 45299 limcresioolb 45300 lptioo2cn 45302 lptioo1cn 45303 limclner 45308 cncfiooicclem1 45550 jumpncnp 45555 dvresioo 45578 dvbdfbdioolem1 45585 itgsin0pilem1 45607 itgsinexplem1 45611 itgsubsticclem 45632 itgiccshift 45637 itgperiod 45638 itgsbtaddcnst 45639 dirkeritg 45759 dirkercncflem2 45761 dirkercncflem3 45762 dirkercncflem4 45763 dirkercncf 45764 fourierdlem28 45792 fourierdlem32 45796 fourierdlem33 45797 fourierdlem39 45803 fourierdlem56 45819 fourierdlem57 45820 fourierdlem58 45821 fourierdlem59 45822 fourierdlem62 45825 fourierdlem68 45831 fourierdlem72 45835 fourierdlem73 45836 fourierdlem74 45837 fourierdlem75 45838 fourierdlem80 45843 fourierdlem94 45857 fourierdlem103 45866 fourierdlem104 45867 fourierdlem113 45876 fouriercnp 45883 fouriersw 45888 fouriercn 45889 etransclem2 45893 etransclem23 45914 etransclem35 45926 etransclem38 45929 etransclem39 45930 etransclem44 45935 etransclem45 45936 etransclem46 45937 etransclem47 45938

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