tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 24839

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 2735	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 24809	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 11210	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 24818	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2735	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 24553	. . 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 24832	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 × cxp 5687 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 − cmin 11490 (,)cioo 13384 abscabs 15270 ↾_t crest 17467 TopOpenctopn 17468 topGenctg 17484 ∞Metcxmet 21367 MetOpencmopn 21372 ℂ_fldccnfld 21382

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-bases 22969

This theorem is referenced by: rerest 24840 tgioo3 24841 zcld2 24851 metdcn 24876 ngnmcncn 24881 metdscn2 24893 abscncfALT 24965 cnrehmeo 24998 cnrehmeoOLD 24999 rellycmp 25003 evth 25005 evth2 25006 lebnumlem2 25008 resscdrg 25406 retopn 25427 cncombf 25707 cnmbf 25708 dvmptresicc 25966 dvcjbr 26002 rolle 26043 cmvth 26044 cmvthOLD 26045 mvth 26046 dvlip 26047 dvlipcn 26048 dvlip2 26049 c1liplem1 26050 dvgt0lem1 26056 dvle 26061 dvivthlem1 26062 dvne0 26065 lhop1lem 26067 lhop2 26069 lhop 26070 dvcnvrelem1 26071 dvcnvrelem2 26072 dvcnvre 26073 dvcvx 26074 dvfsumle 26075 dvfsumleOLD 26076 dvfsumabs 26078 dvfsumlem2 26082 dvfsumlem2OLD 26083 ftc1 26098 ftc1cn 26099 ftc2 26100 ftc2ditglem 26101 itgparts 26103 itgsubstlem 26104 itgpowd 26106 taylthlem2 26431 taylthlem2OLD 26432 efcvx 26508 pige3ALT 26577 dvloglem 26705 logdmopn 26706 advlog 26711 advlogexp 26712 logccv 26720 loglesqrt 26819 lgamgulmlem2 27088 ftalem3 27133 log2sumbnd 27603 nmcnc 30725 ipasslem7 30865 rmulccn 33889 raddcn 33890 ftc2re 34592 knoppcnlem10 36485 knoppcnlem11 36486 broucube 37641 ftc1cnnc 37679 ftc2nc 37689 dvasin 37691 dvacos 37692 dvreasin 37693 dvreacos 37694 areacirclem1 37695 areacirc 37700 dvrelog2 42046 dvrelog3 42047 aks4d1p1p6 42055 redvmptabs 42369 readvrec2 42370 lhe4.4ex1a 44325 refsumcn 44968 xrtgcntopre 45429 tgioo4 45526 climreeq 45569 limcresiooub 45598 limcresioolb 45599 lptioo2cn 45601 lptioo1cn 45602 limclner 45607 cncfiooicclem1 45849 jumpncnp 45854 dvresioo 45877 dvbdfbdioolem1 45884 itgsin0pilem1 45906 itgsinexplem1 45910 itgsubsticclem 45931 itgiccshift 45936 itgperiod 45937 itgsbtaddcnst 45938 dirkeritg 46058 dirkercncflem2 46060 dirkercncflem3 46061 dirkercncflem4 46062 dirkercncf 46063 fourierdlem28 46091 fourierdlem32 46095 fourierdlem33 46096 fourierdlem39 46102 fourierdlem56 46118 fourierdlem57 46119 fourierdlem58 46120 fourierdlem59 46121 fourierdlem62 46124 fourierdlem68 46130 fourierdlem72 46134 fourierdlem73 46135 fourierdlem74 46136 fourierdlem75 46137 fourierdlem80 46142 fourierdlem94 46156 fourierdlem103 46165 fourierdlem104 46166 fourierdlem113 46175 fouriercnp 46182 fouriersw 46187 fouriercn 46188 etransclem2 46192 etransclem23 46213 etransclem35 46225 etransclem38 46228 etransclem39 46229 etransclem44 46234 etransclem45 46235 etransclem46 46236 etransclem47 46237

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