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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 22594 | . . . 4 β’ (π½ β Nrm β π½ β Top) | |
2 | toptopon2 22174 | . . . 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 2736 | . . . 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 23001 | . . 3 β’ ((π½ β (TopOnββͺ π½) β§ π½ β Nrm) β (KQβπ½) β Nrm) |
6 | 3, 5 | mpancom 685 | . 2 β’ (π½ β Nrm β (KQβπ½) β Nrm) |
7 | nrmtop 22594 | . . . . 5 β’ ((KQβπ½) β Nrm β (KQβπ½) β Top) | |
8 | kqtop 23003 | . . . . 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 23002 | . . 3 β’ ((π½ β (TopOnββͺ π½) β§ (KQβπ½) β Nrm) β π½ β Nrm) |
12 | 10, 11 | mpancom 685 | . 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 2105 {crab 3403 βͺ cuni 4853 β¦ cmpt 5176 βcfv 6480 Topctop 22149 TopOnctopon 22166 Nrmcnrm 22568 KQckq 22951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-int 4896 df-iun 4944 df-iin 4945 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-ov 7341 df-oprab 7342 df-mpo 7343 df-map 8689 df-qtop 17316 df-top 22150 df-topon 22167 df-cld 22277 df-cls 22279 df-cn 22485 df-nrm 22575 df-kq 22952 |
This theorem is referenced by: (None) |
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