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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 24781 | . 2 ⊢ ran (,) ∈ TopBases | |
| 2 | tgcl 22976 | . 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 2108 ran crn 5686 ‘cfv 6561 (,)cioo 13387 topGenctg 17482 Topctop 22899 TopBasesctb 22952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-top 22900 df-bases 22953 |
| This theorem is referenced by: retopon 24784 retps 24785 icccld 24787 icopnfcld 24788 iocmnfcld 24789 qdensere 24790 zcld 24835 iccntr 24843 icccmp 24847 reconnlem2 24849 retopconn 24851 rectbntr0 24854 cnmpopc 24955 icoopnst 24969 iocopnst 24970 cnheiborlem 24986 bndth 24990 pcoass 25057 evthicc 25494 ovolicc2 25557 subopnmbl 25639 dvlip 26032 dvlip2 26034 dvne0 26050 lhop2 26054 lhop 26055 dvcnvrelem2 26057 dvcnvre 26058 ftc1 26083 taylthlem2 26416 taylthlem2OLD 26417 cxpcn3 26791 lgamgulmlem2 27073 circtopn 33836 tpr2rico 33911 rrhqima 34015 rrhre 34022 brsiga 34184 unibrsiga 34187 elmbfmvol2 34269 sxbrsigalem3 34274 dya2iocbrsiga 34277 dya2icobrsiga 34278 dya2iocucvr 34286 sxbrsigalem1 34287 orrvcval4 34467 orrvcoel 34468 orrvccel 34469 retopsconn 35254 iccllysconn 35255 rellysconn 35256 cvmliftlem8 35297 cvmliftlem10 35299 ivthALT 36336 ptrecube 37627 poimirlem29 37656 poimirlem30 37657 poimirlem31 37658 poimir 37660 broucube 37661 mblfinlem1 37664 mblfinlem2 37665 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 cnambfre 37675 ftc1cnnc 37699 dvrelog3 42066 redvmptabs 42390 reopn 45301 ioontr 45524 iocopn 45533 icoopn 45538 limciccioolb 45636 limcicciooub 45652 lptre2pt 45655 limcresiooub 45657 limcresioolb 45658 limclner 45666 limclr 45670 icccncfext 45902 cncfiooicclem1 45908 fperdvper 45934 stoweidlem53 46068 stoweidlem57 46072 dirkercncflem2 46119 dirkercncflem3 46120 dirkercncflem4 46121 fourierdlem32 46154 fourierdlem33 46155 fourierdlem42 46164 fourierdlem48 46169 fourierdlem49 46170 fourierdlem58 46179 fourierdlem62 46183 fourierdlem73 46194 fouriersw 46246 iooborel 46366 bor1sal 46370 incsmf 46757 decsmf 46782 smfpimbor1lem2 46814 smf2id 46816 smfco 46817 iooii 48815 |
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