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Theorem kur14lem6 23757
Description: Lemma for kur14 23762. 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 3316 . . . . . . 7  |-  ( X 
\  ( K `  A ) )  C_  X
75, 6eqsstri 3221 . . . . . 6  |-  B  C_  X
81, 2, 3, 4, 7kur14lem3 23754 . . . . 5  |-  ( K `
 B )  C_  X
94fveq1i 5542 . . . . . 6  |-  ( I `
 ( K `  B ) )  =  ( ( int `  J
) `  ( K `  B ) )
102ntrss2 16810 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( K `  B ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  B ) )  C_  ( K `  B ) )
111, 8, 10mp2an 653 . . . . . 6  |-  ( ( int `  J ) `
 ( K `  B ) )  C_  ( K `  B )
129, 11eqsstri 3221 . . . . 5  |-  ( I `
 ( K `  B ) )  C_  ( K `  B )
132clsss 16807 . . . . 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 1277 . . . 4  |-  ( ( cls `  J ) `
 ( I `  ( K `  B ) ) )  C_  (
( cls `  J
) `  ( K `  B ) )
153fveq1i 5542 . . . 4  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
163fveq1i 5542 . . . 4  |-  ( K `
 ( K `  B ) )  =  ( ( cls `  J
) `  ( K `  B ) )
1714, 15, 163sstr4i 3230 . . 3  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  ( K `  B ) )
181, 2, 3, 4, 7kur14lem5 23756 . . 3  |-  ( K `
 ( K `  B ) )  =  ( K `  B
)
1917, 18sseqtri 3223 . 2  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  B )
201, 2, 3, 4, 8kur14lem2 23753 . . . . 5  |-  ( I `
 ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) )
21 difss 3316 . . . . 5  |-  ( X 
\  ( K `  ( X  \  ( K `  B )
) ) )  C_  X
2220, 21eqsstri 3221 . . . 4  |-  ( I `
 ( K `  B ) )  C_  X
23 kur14lem.a . . . . . . . . 9  |-  A  C_  X
241, 2, 3, 4, 23kur14lem3 23754 . . . . . . . 8  |-  ( K `
 A )  C_  X
255fveq2i 5544 . . . . . . . . . . 11  |-  ( K `
 B )  =  ( K `  ( X  \  ( K `  A ) ) )
2625difeq2i 3304 . . . . . . . . . 10  |-  ( X 
\  ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
271, 2, 3, 4, 24kur14lem2 23753 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
284fveq1i 5542 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( ( int `  J
) `  ( K `  A ) )
2926, 27, 283eqtr2i 2322 . . . . . . . . 9  |-  ( X 
\  ( K `  B ) )  =  ( ( int `  J
) `  ( K `  A ) )
302ntrss2 16810 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( K `  A ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  A ) )  C_  ( K `  A ) )
311, 24, 30mp2an 653 . . . . . . . . 9  |-  ( ( int `  J ) `
 ( K `  A ) )  C_  ( K `  A )
3229, 31eqsstri 3221 . . . . . . . 8  |-  ( X 
\  ( K `  B ) )  C_  ( K `  A )
332clsss 16807 . . . . . . . 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 1277 . . . . . . 7  |-  ( ( cls `  J ) `
 ( X  \ 
( K `  B
) ) )  C_  ( ( cls `  J
) `  ( K `  A ) )
353fveq1i 5542 . . . . . . 7  |-  ( K `
 ( X  \ 
( K `  B
) ) )  =  ( ( cls `  J
) `  ( X  \  ( K `  B
) ) )
361, 2, 3, 4, 23kur14lem5 23756 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)
373fveq1i 5542 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( ( cls `  J
) `  ( K `  A ) )
3836, 37eqtr3i 2318 . . . . . . 7  |-  ( K `
 A )  =  ( ( cls `  J
) `  ( K `  A ) )
3934, 35, 383sstr4i 3230 . . . . . 6  |-  ( K `
 ( X  \ 
( K `  B
) ) )  C_  ( K `  A )
40 sscon 3323 . . . . . 6  |-  ( ( K `  ( X 
\  ( K `  B ) ) ) 
C_  ( K `  A )  ->  ( X  \  ( K `  A ) )  C_  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) ) )
4139, 40ax-mp 8 . . . . 5  |-  ( X 
\  ( K `  A ) )  C_  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) )
4241, 5, 203sstr4i 3230 . . . 4  |-  B  C_  ( I `  ( K `  B )
)
432clsss 16807 . . . 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 1277 . . 3  |-  ( ( cls `  J ) `
 B )  C_  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
453fveq1i 5542 . . 3  |-  ( K `
 B )  =  ( ( cls `  J
) `  B )
4644, 45, 153sstr4i 3230 . 2  |-  ( K `
 B )  C_  ( K `  ( I `
 ( K `  B ) ) )
4719, 46eqssi 3208 1  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( K `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   intcnt 16770   clsccl 16771
This theorem is referenced by:  kur14lem7  23758
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-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774
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