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Mirrors > Home > MPE Home > Th. List > retop | Structured version Visualization version GIF version |
Description: The standard topology on the reals. (Contributed by FL, 4-Jun-2007.) |
Ref | Expression |
---|---|
retop | ⊢ (topGen‘ran (,)) ∈ Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retopbas 24768 | . 2 ⊢ ran (,) ∈ TopBases | |
2 | tgcl 22963 | . 2 ⊢ (ran (,) ∈ TopBases → (topGen‘ran (,)) ∈ Top) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (topGen‘ran (,)) ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ran crn 5683 ‘cfv 6554 (,)cioo 13378 topGenctg 17452 Topctop 22886 TopBasesctb 22939 |
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-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
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-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-ioo 13382 df-topgen 17458 df-top 22887 df-bases 22940 |
This theorem is referenced by: retopon 24771 retps 24772 icccld 24774 icopnfcld 24775 iocmnfcld 24776 qdensere 24777 zcld 24820 iccntr 24828 icccmp 24832 reconnlem2 24834 retopconn 24836 rectbntr0 24839 cnmpopc 24940 icoopnst 24954 iocopnst 24955 cnheiborlem 24971 bndth 24975 pcoass 25042 evthicc 25479 ovolicc2 25542 subopnmbl 25624 dvlip 26017 dvlip2 26019 dvne0 26035 lhop2 26039 lhop 26040 dvcnvrelem2 26042 dvcnvre 26043 ftc1 26068 taylthlem2 26402 taylthlem2OLD 26403 cxpcn3 26776 lgamgulmlem2 27058 circtopn 33652 tpr2rico 33727 rrhqima 33829 rrhre 33836 brsiga 34016 unibrsiga 34019 elmbfmvol2 34101 sxbrsigalem3 34106 dya2iocbrsiga 34109 dya2icobrsiga 34110 dya2iocucvr 34118 sxbrsigalem1 34119 orrvcval4 34298 orrvcoel 34299 orrvccel 34300 retopsconn 35077 iccllysconn 35078 rellysconn 35079 cvmliftlem8 35120 cvmliftlem10 35122 ivthALT 36047 ptrecube 37321 poimirlem29 37350 poimirlem30 37351 poimirlem31 37352 poimir 37354 broucube 37355 mblfinlem1 37358 mblfinlem2 37359 mblfinlem3 37360 mblfinlem4 37361 ismblfin 37362 cnambfre 37369 ftc1cnnc 37393 dvrelog3 41764 reopn 44904 ioontr 45129 iocopn 45138 icoopn 45143 limciccioolb 45242 limcicciooub 45258 lptre2pt 45261 limcresiooub 45263 limcresioolb 45264 limclner 45272 limclr 45276 icccncfext 45508 cncfiooicclem1 45514 fperdvper 45540 stoweidlem53 45674 stoweidlem57 45678 dirkercncflem2 45725 dirkercncflem3 45726 dirkercncflem4 45727 fourierdlem32 45760 fourierdlem33 45761 fourierdlem42 45770 fourierdlem48 45775 fourierdlem49 45776 fourierdlem58 45785 fourierdlem62 45789 fourierdlem73 45800 fouriersw 45852 iooborel 45972 bor1sal 45976 incsmf 46363 decsmf 46388 smfpimbor1lem2 46420 smf2id 46422 smfco 46423 iooii 48251 |
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