tgioo2 - Metamath Proof Explorer

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Theorem tgioo2 24706

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 2727	. 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
2		cnxmet 24676	. . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
3		ax-resscn 11187	. . 3 ⊢ ℝ ⊆ ℂ
4		tgioo2.1	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
5	4	cnfldtopn 24685	. . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
6		eqid 2727	. . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
7	1, 5, 6	metrest 24420	. . 3 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ ℝ ⊆ ℂ) → (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))))
8	2, 3, 7	mp2an 691	. 2 ⊢ (𝐽 ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
9	1, 8	tgioo 24699	1 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 × cxp 5670 ran crn 5673 ↾ cres 5674 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 ℝcr 11129 − cmin 11466 (,)cioo 13348 abscabs 15205 ↾_t crest 17393 TopOpenctopn 17394 topGenctg 17410 ∞Metcxmet 21251 MetOpencmopn 21256 ℂ_fldccnfld 21266

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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-fz 13509 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-struct 17107 df-slot 17142 df-ndx 17154 df-base 17172 df-plusg 17237 df-mulr 17238 df-starv 17239 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-rest 17395 df-topn 17396 df-topgen 17416 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-cnfld 21267 df-top 22783 df-topon 22800 df-bases 22836

This theorem is referenced by: rerest 24707 tgioo3 24708 zcld2 24718 metdcn 24743 ngnmcncn 24748 metdscn2 24760 abscncfALT 24832 cnrehmeo 24865 cnrehmeoOLD 24866 rellycmp 24870 evth 24872 evth2 24873 lebnumlem2 24875 resscdrg 25273 retopn 25294 cncombf 25574 cnmbf 25575 dvmptresicc 25832 dvcjbr 25868 rolle 25909 cmvth 25910 cmvthOLD 25911 mvth 25912 dvlip 25913 dvlipcn 25914 dvlip2 25915 c1liplem1 25916 dvgt0lem1 25922 dvle 25927 dvivthlem1 25928 dvne0 25931 lhop1lem 25933 lhop2 25935 lhop 25936 dvcnvrelem1 25937 dvcnvrelem2 25938 dvcnvre 25939 dvcvx 25940 dvfsumle 25941 dvfsumleOLD 25942 dvfsumabs 25944 dvfsumlem2 25948 dvfsumlem2OLD 25949 ftc1 25964 ftc1cn 25965 ftc2 25966 ftc2ditglem 25967 itgparts 25969 itgsubstlem 25970 itgpowd 25972 taylthlem2 26296 taylthlem2OLD 26297 efcvx 26373 pige3ALT 26441 dvloglem 26569 logdmopn 26570 advlog 26575 advlogexp 26576 logccv 26584 loglesqrt 26680 lgamgulmlem2 26949 ftalem3 26994 log2sumbnd 27464 nmcnc 30493 ipasslem7 30633 rmulccn 33465 raddcn 33466 ftc2re 34166 knoppcnlem10 35913 knoppcnlem11 35914 broucube 37062 ftc1cnnc 37100 ftc2nc 37110 dvasin 37112 dvacos 37113 dvreasin 37114 dvreacos 37115 areacirclem1 37116 areacirc 37121 dvrelog2 41472 dvrelog3 41473 aks4d1p1p6 41481 lhe4.4ex1a 43689 refsumcn 44315 xrtgcntopre 44784 tgioo4 44881 climreeq 44924 limcresiooub 44953 limcresioolb 44954 lptioo2cn 44956 lptioo1cn 44957 limclner 44962 cncfiooicclem1 45204 jumpncnp 45209 dvresioo 45232 dvbdfbdioolem1 45239 itgsin0pilem1 45261 itgsinexplem1 45265 itgsubsticclem 45286 itgiccshift 45291 itgperiod 45292 itgsbtaddcnst 45293 dirkeritg 45413 dirkercncflem2 45415 dirkercncflem3 45416 dirkercncflem4 45417 dirkercncf 45418 fourierdlem28 45446 fourierdlem32 45450 fourierdlem33 45451 fourierdlem39 45457 fourierdlem56 45473 fourierdlem57 45474 fourierdlem58 45475 fourierdlem59 45476 fourierdlem62 45479 fourierdlem68 45485 fourierdlem72 45489 fourierdlem73 45490 fourierdlem74 45491 fourierdlem75 45492 fourierdlem80 45497 fourierdlem94 45511 fourierdlem103 45520 fourierdlem104 45521 fourierdlem113 45530 fouriercnp 45537 fouriersw 45542 fouriercn 45543 etransclem2 45547 etransclem23 45568 etransclem35 45580 etransclem38 45583 etransclem39 45584 etransclem44 45589 etransclem45 45590 etransclem46 45591 etransclem47 45592

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