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Theorem ist1-5lem 17527
Description: Lemma for ist1-5 17529 and similar theorems. If  A 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  A (which is defined as stating that the Kolmogorov quotient of the space has property  A). For example, if  A 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  |-  ( J  e.  A  ->  J  e.  Kol2 )
ist1-5lem.2  |-  ( J  ~=  (KQ `  J
)  ->  ( J  e.  A  ->  (KQ `  J )  e.  A
) )
ist1-5lem.3  |-  ( (KQ
`  J )  ~=  J  ->  ( (KQ `  J )  e.  A  ->  J  e.  A ) )
Assertion
Ref Expression
ist1-5lem  |-  ( J  e.  A  <->  ( J  e.  Kol2  /\  (KQ `  J
)  e.  A ) )

Proof of Theorem ist1-5lem
StepHypRef Expression
1 ist1-5lem.1 . . 3  |-  ( J  e.  A  ->  J  e.  Kol2 )
2 kqhmph 17526 . . . . 5  |-  ( J  e.  Kol2  <->  J  ~=  (KQ `  J ) )
31, 2sylib 188 . . . 4  |-  ( J  e.  A  ->  J  ~=  (KQ `  J ) )
4 ist1-5lem.2 . . . 4  |-  ( J  ~=  (KQ `  J
)  ->  ( J  e.  A  ->  (KQ `  J )  e.  A
) )
53, 4mpcom 32 . . 3  |-  ( J  e.  A  ->  (KQ `  J )  e.  A
)
61, 5jca 518 . 2  |-  ( J  e.  A  ->  ( J  e.  Kol2  /\  (KQ `  J )  e.  A
) )
7 hmphsym 17489 . . . . 5  |-  ( J  ~=  (KQ `  J
)  ->  (KQ `  J
)  ~=  J )
82, 7sylbi 187 . . . 4  |-  ( J  e.  Kol2  ->  (KQ `  J )  ~=  J
)
9 ist1-5lem.3 . . . 4  |-  ( (KQ
`  J )  ~=  J  ->  ( (KQ `  J )  e.  A  ->  J  e.  A ) )
108, 9syl 15 . . 3  |-  ( J  e.  Kol2  ->  ( (KQ
`  J )  e.  A  ->  J  e.  A ) )
1110imp 418 . 2  |-  ( ( J  e.  Kol2  /\  (KQ `  J )  e.  A
)  ->  J  e.  A )
126, 11impbii 180 1  |-  ( J  e.  A  <->  ( J  e.  Kol2  /\  (KQ `  J
)  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Kol2ct0 17050  KQckq 17400    ~= chmph 17461
This theorem is referenced by:  ist1-5  17529  ishaus3  17530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-map 6790  df-qtop 13426  df-top 16652  df-topon 16655  df-cn 16973  df-t0 17057  df-kq 17401  df-hmeo 17462  df-hmph 17463
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