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Mirrors > Home > MPE Home > Th. List > t0kq | Structured version Visualization version Unicode version |
Description: A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
t0kq.1 |
Ref | Expression |
---|---|
t0kq | TopOn KQ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . 4 TopOn TopOn | |
2 | t0kq.1 | . . . . . 6 | |
3 | 2 | ist0-4 21532 | . . . . 5 TopOn |
4 | 3 | biimpa 501 | . . . 4 TopOn |
5 | 1, 4 | qtopf1 21619 | . . 3 TopOn qTop |
6 | 2 | kqval 21529 | . . . . 5 TopOn KQ qTop |
7 | 6 | adantr 481 | . . . 4 TopOn KQ qTop |
8 | 7 | oveq2d 6666 | . . 3 TopOn KQ qTop |
9 | 5, 8 | eleqtrrd 2704 | . 2 TopOn KQ |
10 | hmphi 21580 | . . . . 5 KQ KQ | |
11 | hmphsym 21585 | . . . . 5 KQ KQ | |
12 | 10, 11 | syl 17 | . . . 4 KQ KQ |
13 | 2 | kqt0lem 21539 | . . . 4 TopOn KQ |
14 | t0hmph 21593 | . . . 4 KQ KQ | |
15 | 12, 13, 14 | syl2im 40 | . . 3 KQ TopOn |
16 | 15 | impcom 446 | . 2 TopOn KQ |
17 | 9, 16 | impbida 877 | 1 TopOn KQ |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 class class class wbr 4653 cmpt 4729 wf1 5885 cfv 5888 (class class class)co 6650 qTop cqtop 16163 TopOnctopon 20715 ct0 21110 KQckq 21496 chmeo 21556 chmph 21557 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-reu 2919 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-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 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-1o 7560 df-map 7859 df-qtop 16167 df-top 20699 df-topon 20716 df-cn 21031 df-t0 21117 df-kq 21497 df-hmeo 21558 df-hmph 21559 |
This theorem is referenced by: kqhmph 21622 |
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