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Mirrors > Home > MPE Home > Th. List > kqnrm | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqnrm | β’ (π½ β Nrm β (KQβπ½) β Nrm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrmtop 23239 | . . . 4 β’ (π½ β Nrm β π½ β Top) | |
2 | toptopon2 22819 | . . . 4 β’ (π½ β Top β π½ β (TopOnββͺ π½)) | |
3 | 1, 2 | sylib 217 | . . 3 β’ (π½ β Nrm β π½ β (TopOnββͺ π½)) |
4 | eqid 2728 | . . . 4 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 | kqnrmlem1 23646 | . . 3 β’ ((π½ β (TopOnββͺ π½) β§ π½ β Nrm) β (KQβπ½) β Nrm) |
6 | 3, 5 | mpancom 687 | . 2 β’ (π½ β Nrm β (KQβπ½) β Nrm) |
7 | nrmtop 23239 | . . . . 5 β’ ((KQβπ½) β Nrm β (KQβπ½) β Top) | |
8 | kqtop 23648 | . . . . 5 β’ (π½ β Top β (KQβπ½) β Top) | |
9 | 7, 8 | sylibr 233 | . . . 4 β’ ((KQβπ½) β Nrm β π½ β Top) |
10 | 9, 2 | sylib 217 | . . 3 β’ ((KQβπ½) β Nrm β π½ β (TopOnββͺ π½)) |
11 | 4 | kqnrmlem2 23647 | . . 3 β’ ((π½ β (TopOnββͺ π½) β§ (KQβπ½) β Nrm) β π½ β Nrm) |
12 | 10, 11 | mpancom 687 | . 2 β’ ((KQβπ½) β Nrm β π½ β Nrm) |
13 | 6, 12 | impbii 208 | 1 β’ (π½ β Nrm β (KQβπ½) β Nrm) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β wcel 2099 {crab 3429 βͺ cuni 4908 β¦ cmpt 5231 βcfv 6548 Topctop 22794 TopOnctopon 22811 Nrmcnrm 23213 KQckq 23596 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-qtop 17488 df-top 22795 df-topon 22812 df-cld 22922 df-cls 22924 df-cn 23130 df-nrm 23220 df-kq 23597 |
This theorem is referenced by: (None) |
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