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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 22837 | . . 3 β’ (π½ β Reg β π½ β Top) | |
2 | toptopon2 22420 | . . 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 2733 | . . 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 23244 | . 2 β’ ((π½ β (TopOnββͺ π½) β§ π½ β Reg) β (KQβπ½) β Haus) |
6 | 3, 5 | mpancom 687 | 1 β’ (π½ β Reg β (KQβπ½) β Haus) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2107 {crab 3433 βͺ cuni 4909 β¦ cmpt 5232 βcfv 6544 Topctop 22395 TopOnctopon 22412 Hauscha 22812 Regcreg 22813 KQckq 23197 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-qtop 17453 df-top 22396 df-topon 22413 df-cld 22523 df-cls 22525 df-haus 22819 df-reg 22820 df-kq 23198 |
This theorem is referenced by: reghaus 23329 |
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