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Theorem kur14lem6 29508
Description: Lemma for kur14 29513. If  k is the complementation operator and  k is the closure operator, this expresses the identity  k c
k A  =  k c k c k c k A for any subset  A of the topological space. This is the key result that lets us cut down long enough sequences of  c k c k ... that arise when applying closure and complement repeatedly to  A, and explains why we end up with a number as large as  1 4, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j  |-  J  e. 
Top
kur14lem.x  |-  X  = 
U. J
kur14lem.k  |-  K  =  ( cls `  J
)
kur14lem.i  |-  I  =  ( int `  J
)
kur14lem.a  |-  A  C_  X
kur14lem.b  |-  B  =  ( X  \  ( K `  A )
)
Assertion
Ref Expression
kur14lem6  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( K `  B )

Proof of Theorem kur14lem6
StepHypRef Expression
1 kur14lem.j . . . . 5  |-  J  e. 
Top
2 kur14lem.x . . . . . 6  |-  X  = 
U. J
3 kur14lem.k . . . . . 6  |-  K  =  ( cls `  J
)
4 kur14lem.i . . . . . 6  |-  I  =  ( int `  J
)
5 kur14lem.b . . . . . . 7  |-  B  =  ( X  \  ( K `  A )
)
6 difss 3570 . . . . . . 7  |-  ( X 
\  ( K `  A ) )  C_  X
75, 6eqsstri 3472 . . . . . 6  |-  B  C_  X
81, 2, 3, 4, 7kur14lem3 29505 . . . . 5  |-  ( K `
 B )  C_  X
94fveq1i 5850 . . . . . 6  |-  ( I `
 ( K `  B ) )  =  ( ( int `  J
) `  ( K `  B ) )
102ntrss2 19850 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( K `  B ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  B ) )  C_  ( K `  B ) )
111, 8, 10mp2an 670 . . . . . 6  |-  ( ( int `  J ) `
 ( K `  B ) )  C_  ( K `  B )
129, 11eqsstri 3472 . . . . 5  |-  ( I `
 ( K `  B ) )  C_  ( K `  B )
132clsss 19847 . . . . 5  |-  ( ( J  e.  Top  /\  ( K `  B ) 
C_  X  /\  (
I `  ( K `  B ) )  C_  ( K `  B ) )  ->  ( ( cls `  J ) `  ( I `  ( K `  B )
) )  C_  (
( cls `  J
) `  ( K `  B ) ) )
141, 8, 12, 13mp3an 1326 . . . 4  |-  ( ( cls `  J ) `
 ( I `  ( K `  B ) ) )  C_  (
( cls `  J
) `  ( K `  B ) )
153fveq1i 5850 . . . 4  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
163fveq1i 5850 . . . 4  |-  ( K `
 ( K `  B ) )  =  ( ( cls `  J
) `  ( K `  B ) )
1714, 15, 163sstr4i 3481 . . 3  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  ( K `  B ) )
181, 2, 3, 4, 7kur14lem5 29507 . . 3  |-  ( K `
 ( K `  B ) )  =  ( K `  B
)
1917, 18sseqtri 3474 . 2  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  B )
201, 2, 3, 4, 8kur14lem2 29504 . . . . 5  |-  ( I `
 ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) )
21 difss 3570 . . . . 5  |-  ( X 
\  ( K `  ( X  \  ( K `  B )
) ) )  C_  X
2220, 21eqsstri 3472 . . . 4  |-  ( I `
 ( K `  B ) )  C_  X
23 kur14lem.a . . . . . . . . 9  |-  A  C_  X
241, 2, 3, 4, 23kur14lem3 29505 . . . . . . . 8  |-  ( K `
 A )  C_  X
255fveq2i 5852 . . . . . . . . . . 11  |-  ( K `
 B )  =  ( K `  ( X  \  ( K `  A ) ) )
2625difeq2i 3558 . . . . . . . . . 10  |-  ( X 
\  ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
271, 2, 3, 4, 24kur14lem2 29504 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
284fveq1i 5850 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( ( int `  J
) `  ( K `  A ) )
2926, 27, 283eqtr2i 2437 . . . . . . . . 9  |-  ( X 
\  ( K `  B ) )  =  ( ( int `  J
) `  ( K `  A ) )
302ntrss2 19850 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( K `  A ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  A ) )  C_  ( K `  A ) )
311, 24, 30mp2an 670 . . . . . . . . 9  |-  ( ( int `  J ) `
 ( K `  A ) )  C_  ( K `  A )
3229, 31eqsstri 3472 . . . . . . . 8  |-  ( X 
\  ( K `  B ) )  C_  ( K `  A )
332clsss 19847 . . . . . . . 8  |-  ( ( J  e.  Top  /\  ( K `  A ) 
C_  X  /\  ( X  \  ( K `  B ) )  C_  ( K `  A ) )  ->  ( ( cls `  J ) `  ( X  \  ( K `  B )
) )  C_  (
( cls `  J
) `  ( K `  A ) ) )
341, 24, 32, 33mp3an 1326 . . . . . . 7  |-  ( ( cls `  J ) `
 ( X  \ 
( K `  B
) ) )  C_  ( ( cls `  J
) `  ( K `  A ) )
353fveq1i 5850 . . . . . . 7  |-  ( K `
 ( X  \ 
( K `  B
) ) )  =  ( ( cls `  J
) `  ( X  \  ( K `  B
) ) )
361, 2, 3, 4, 23kur14lem5 29507 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)
373fveq1i 5850 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( ( cls `  J
) `  ( K `  A ) )
3836, 37eqtr3i 2433 . . . . . . 7  |-  ( K `
 A )  =  ( ( cls `  J
) `  ( K `  A ) )
3934, 35, 383sstr4i 3481 . . . . . 6  |-  ( K `
 ( X  \ 
( K `  B
) ) )  C_  ( K `  A )
40 sscon 3577 . . . . . 6  |-  ( ( K `  ( X 
\  ( K `  B ) ) ) 
C_  ( K `  A )  ->  ( X  \  ( K `  A ) )  C_  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) ) )
4139, 40ax-mp 5 . . . . 5  |-  ( X 
\  ( K `  A ) )  C_  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) )
4241, 5, 203sstr4i 3481 . . . 4  |-  B  C_  ( I `  ( K `  B )
)
432clsss 19847 . . . 4  |-  ( ( J  e.  Top  /\  ( I `  ( K `  B )
)  C_  X  /\  B  C_  ( I `  ( K `  B ) ) )  ->  (
( cls `  J
) `  B )  C_  ( ( cls `  J
) `  ( I `  ( K `  B
) ) ) )
441, 22, 42, 43mp3an 1326 . . 3  |-  ( ( cls `  J ) `
 B )  C_  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
453fveq1i 5850 . . 3  |-  ( K `
 B )  =  ( ( cls `  J
) `  B )
4644, 45, 153sstr4i 3481 . 2  |-  ( K `
 B )  C_  ( K `  ( I `
 ( K `  B ) ) )
4719, 46eqssi 3458 1  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( K `  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842    \ cdif 3411    C_ wss 3414   U.cuni 4191   ` cfv 5569   Topctop 19686   intcnt 19810   clsccl 19811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-top 19691  df-cld 19812  df-ntr 19813  df-cls 19814
This theorem is referenced by:  kur14lem7  29509
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