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Theorem ist1-5lem 17511
Description: Lemma for ist1-5 17513 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 17510 . . . . 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 17473 . . . . 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 1684   class class class wbr 4023   ` cfv 5255   Kol2ct0 17034  KQckq 17384    ~= chmph 17445
This theorem is referenced by:  ist1-5  17513  ishaus3  17514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cn 16957  df-t0 17041  df-kq 17385  df-hmeo 17446  df-hmph 17447
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