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Theorem ist1-5lem 17842
 Description: Lemma for ist1-5 17844 and similar theorems. If is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property (which is defined as stating that the Kolmogorov quotient of the space has property ). For example, if is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
ist1-5lem.1
ist1-5lem.2 KQ KQ
ist1-5lem.3 KQ KQ
Assertion
Ref Expression
ist1-5lem KQ

Proof of Theorem ist1-5lem
StepHypRef Expression
1 ist1-5lem.1 . . 3
2 kqhmph 17841 . . . . 5 KQ
31, 2sylib 189 . . . 4 KQ
4 ist1-5lem.2 . . . 4 KQ KQ
53, 4mpcom 34 . . 3 KQ
61, 5jca 519 . 2 KQ
7 hmphsym 17804 . . . . 5 KQ KQ
82, 7sylbi 188 . . . 4 KQ
9 ist1-5lem.3 . . . 4 KQ KQ
108, 9syl 16 . . 3 KQ
1110imp 419 . 2 KQ
126, 11impbii 181 1 KQ
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wcel 1725   class class class wbr 4204  cfv 5446  ct0 17360  KQckq 17715   chmph 17776 This theorem is referenced by:  ist1-5  17844  ishaus3  17845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-map 7012  df-qtop 13723  df-top 16953  df-topon 16956  df-cn 17281  df-t0 17367  df-kq 17716  df-hmeo 17777  df-hmph 17778
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