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Mirrors > Home > MPE Home > Th. List > regr1 | Structured version Visualization version GIF version |
Description: A regular space is R1, which means that any two topologically distinct points can be separated by neighborhoods. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
regr1 | β’ (π½ β Reg β (KQβπ½) β Haus) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | regtop 22836 | . . 3 β’ (π½ β Reg β π½ β Top) | |
2 | toptopon2 22419 | . . 3 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
3 | 1, 2 | sylib 217 | . 2 β’ (π½ β Reg β π½ β (TopOnββͺ π½)) |
4 | eqid 2732 | . . 3 β’ (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦}) = (π₯ β βͺ π½ β¦ {π¦ β π½ β£ π₯ β π¦}) | |
5 | 4 | regr1lem2 23243 | . 2 β’ ((π½ β (TopOnββͺ π½) β§ π½ β Reg) β (KQβπ½) β Haus) |
6 | 3, 5 | mpancom 686 | 1 β’ (π½ β Reg β (KQβπ½) β Haus) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2106 {crab 3432 βͺ cuni 4908 β¦ cmpt 5231 βcfv 6543 Topctop 22394 TopOnctopon 22411 Hauscha 22811 Regcreg 22812 KQckq 23196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-qtop 17452 df-top 22395 df-topon 22412 df-cld 22522 df-cls 22524 df-haus 22818 df-reg 22819 df-kq 23197 |
This theorem is referenced by: reghaus 23328 |
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