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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 22564 | . 2 ⊢ ran (,) ∈ TopBases | |
2 | tgcl 20773 | . 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 1990 ran crn 5115 ‘cfv 5888 (,)cioo 12175 topGenctg 16098 Topctop 20698 TopBasesctb 20749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-topgen 16104 df-top 20699 df-bases 20750 |
This theorem is referenced by: retopon 22567 retps 22568 icccld 22570 icopnfcld 22571 iocmnfcld 22572 qdensere 22573 zcld 22616 iccntr 22624 icccmp 22628 reconnlem2 22630 retopconn 22632 rectbntr0 22635 cnmpt2pc 22727 icoopnst 22738 iocopnst 22739 cnheiborlem 22753 bndth 22757 pcoass 22824 evthicc 23228 ovolicc2 23290 subopnmbl 23372 dvlip 23756 dvlip2 23758 dvne0 23774 lhop2 23778 lhop 23779 dvcnvrelem2 23781 dvcnvre 23782 ftc1 23805 taylthlem2 24128 cxpcn3 24489 lgamgulmlem2 24756 circtopn 29904 tpr2rico 29958 rrhqima 30058 rrhre 30065 brsiga 30246 unibrsiga 30249 elmbfmvol2 30329 sxbrsigalem3 30334 dya2iocbrsiga 30337 dya2icobrsiga 30338 dya2iocucvr 30346 sxbrsigalem1 30347 orrvcval4 30526 orrvcoel 30527 orrvccel 30528 retopsconn 31231 iccllysconn 31232 rellysconn 31233 cvmliftlem8 31274 cvmliftlem10 31276 ivthALT 32330 ptrecube 33409 poimirlem29 33438 poimirlem30 33439 poimirlem31 33440 poimir 33442 broucube 33443 mblfinlem1 33446 mblfinlem2 33447 mblfinlem3 33448 mblfinlem4 33449 ismblfin 33450 cnambfre 33458 ftc1cnnc 33484 reopn 39501 ioontr 39736 iocopn 39746 icoopn 39751 limciccioolb 39853 limcicciooub 39869 lptre2pt 39872 limcresiooub 39874 limcresioolb 39875 limclner 39883 limclr 39887 icccncfext 40100 cncfiooicclem1 40106 fperdvper 40133 stoweidlem53 40270 stoweidlem57 40274 dirkercncflem2 40321 dirkercncflem3 40322 dirkercncflem4 40323 fourierdlem32 40356 fourierdlem33 40357 fourierdlem42 40366 fourierdlem48 40371 fourierdlem49 40372 fourierdlem58 40381 fourierdlem62 40385 fourierdlem73 40396 fouriersw 40448 iooborel 40569 bor1sal 40573 incsmf 40951 decsmf 40975 smfpimbor1lem2 41006 smf2id 41008 smfco 41009 |
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