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Mirrors > Home > MPE Home > Th. List > tgioo2 | Structured version Visualization version GIF version |
Description: The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
tgioo2.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
tgioo2 | ⊢ (topGen‘ran (,)) = (𝐽 ↾t ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
2 | cnxmet 24281 | . . 3 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
3 | ax-resscn 11164 | . . 3 ⊢ ℝ ⊆ ℂ | |
4 | tgioo2.1 | . . . . 5 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
5 | 4 | cnfldtopn 24290 | . . . 4 ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) |
6 | eqid 2733 | . . . 4 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) | |
7 | 1, 5, 6 | metrest 24025 | . . 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 24304 | 1 ⊢ (topGen‘ran (,)) = (𝐽 ↾t ℝ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 × cxp 5674 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 − cmin 11441 (,)cioo 13321 abscabs 15178 ↾t crest 17363 TopOpenctopn 17364 topGenctg 17380 ∞Metcxmet 20922 MetOpencmopn 20927 ℂfldccnfld 20937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-fz 13482 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-rest 17365 df-topn 17366 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-cnfld 20938 df-top 22388 df-topon 22405 df-bases 22441 |
This theorem is referenced by: rerest 24312 tgioo3 24313 zcld2 24323 metdcn 24348 ngnmcncn 24353 metdscn2 24365 abscncfALT 24432 cnrehmeo 24461 rellycmp 24465 evth 24467 evth2 24468 lebnumlem2 24470 resscdrg 24867 retopn 24888 cncombf 25167 cnmbf 25168 dvmptresicc 25425 dvcjbr 25458 rolle 25499 cmvth 25500 mvth 25501 dvlip 25502 dvlipcn 25503 dvlip2 25504 c1liplem1 25505 dvgt0lem1 25511 dvle 25516 dvivthlem1 25517 dvne0 25520 lhop1lem 25522 lhop2 25524 lhop 25525 dvcnvrelem1 25526 dvcnvrelem2 25527 dvcnvre 25528 dvcvx 25529 dvfsumle 25530 dvfsumabs 25532 dvfsumlem2 25536 ftc1 25551 ftc1cn 25552 ftc2 25553 ftc2ditglem 25554 itgparts 25556 itgsubstlem 25557 itgpowd 25559 taylthlem2 25878 efcvx 25953 pige3ALT 26021 dvloglem 26148 logdmopn 26149 advlog 26154 advlogexp 26155 logccv 26163 loglesqrt 26256 lgamgulmlem2 26524 ftalem3 26569 log2sumbnd 27037 nmcnc 29937 ipasslem7 30077 rmulccn 32897 raddcn 32898 ftc2re 33599 gg-cnrehmeo 35160 gg-rmulccn 35168 gg-cmvth 35170 gg-dvfsumle 35171 gg-dvfsumlem2 35172 knoppcnlem10 35367 knoppcnlem11 35368 broucube 36511 ftc1cnnc 36549 ftc2nc 36559 dvasin 36561 dvacos 36562 dvreasin 36563 dvreacos 36564 areacirclem1 36565 areacirc 36570 dvrelog2 40918 dvrelog3 40919 aks4d1p1p6 40927 lhe4.4ex1a 43074 refsumcn 43700 xrtgcntopre 44176 tgioo4 44273 climreeq 44316 limcresiooub 44345 limcresioolb 44346 lptioo2cn 44348 lptioo1cn 44349 limclner 44354 cncfiooicclem1 44596 jumpncnp 44601 dvresioo 44624 dvbdfbdioolem1 44631 itgsin0pilem1 44653 itgsinexplem1 44657 itgsubsticclem 44678 itgiccshift 44683 itgperiod 44684 itgsbtaddcnst 44685 dirkeritg 44805 dirkercncflem2 44807 dirkercncflem3 44808 dirkercncflem4 44809 dirkercncf 44810 fourierdlem28 44838 fourierdlem32 44842 fourierdlem33 44843 fourierdlem39 44849 fourierdlem56 44865 fourierdlem57 44866 fourierdlem58 44867 fourierdlem59 44868 fourierdlem62 44871 fourierdlem68 44877 fourierdlem72 44881 fourierdlem73 44882 fourierdlem74 44883 fourierdlem75 44884 fourierdlem80 44889 fourierdlem94 44903 fourierdlem103 44912 fourierdlem104 44913 fourierdlem113 44922 fouriercnp 44929 fouriersw 44934 fouriercn 44935 etransclem2 44939 etransclem23 44960 etransclem35 44972 etransclem38 44975 etransclem39 44976 etransclem44 44981 etransclem45 44982 etransclem46 44983 etransclem47 44984 |
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