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Theorem kur14lem6 24889
Description: Lemma for kur14 24894. 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 3466 . . . . . . 7  |-  ( X 
\  ( K `  A ) )  C_  X
75, 6eqsstri 3370 . . . . . 6  |-  B  C_  X
81, 2, 3, 4, 7kur14lem3 24886 . . . . 5  |-  ( K `
 B )  C_  X
94fveq1i 5721 . . . . . 6  |-  ( I `
 ( K `  B ) )  =  ( ( int `  J
) `  ( K `  B ) )
102ntrss2 17113 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( K `  B ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  B ) )  C_  ( K `  B ) )
111, 8, 10mp2an 654 . . . . . 6  |-  ( ( int `  J ) `
 ( K `  B ) )  C_  ( K `  B )
129, 11eqsstri 3370 . . . . 5  |-  ( I `
 ( K `  B ) )  C_  ( K `  B )
132clsss 17110 . . . . 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 1279 . . . 4  |-  ( ( cls `  J ) `
 ( I `  ( K `  B ) ) )  C_  (
( cls `  J
) `  ( K `  B ) )
153fveq1i 5721 . . . 4  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
163fveq1i 5721 . . . 4  |-  ( K `
 ( K `  B ) )  =  ( ( cls `  J
) `  ( K `  B ) )
1714, 15, 163sstr4i 3379 . . 3  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  ( K `  B ) )
181, 2, 3, 4, 7kur14lem5 24888 . . 3  |-  ( K `
 ( K `  B ) )  =  ( K `  B
)
1917, 18sseqtri 3372 . 2  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  B )
201, 2, 3, 4, 8kur14lem2 24885 . . . . 5  |-  ( I `
 ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) )
21 difss 3466 . . . . 5  |-  ( X 
\  ( K `  ( X  \  ( K `  B )
) ) )  C_  X
2220, 21eqsstri 3370 . . . 4  |-  ( I `
 ( K `  B ) )  C_  X
23 kur14lem.a . . . . . . . . 9  |-  A  C_  X
241, 2, 3, 4, 23kur14lem3 24886 . . . . . . . 8  |-  ( K `
 A )  C_  X
255fveq2i 5723 . . . . . . . . . . 11  |-  ( K `
 B )  =  ( K `  ( X  \  ( K `  A ) ) )
2625difeq2i 3454 . . . . . . . . . 10  |-  ( X 
\  ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
271, 2, 3, 4, 24kur14lem2 24885 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
284fveq1i 5721 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( ( int `  J
) `  ( K `  A ) )
2926, 27, 283eqtr2i 2461 . . . . . . . . 9  |-  ( X 
\  ( K `  B ) )  =  ( ( int `  J
) `  ( K `  A ) )
302ntrss2 17113 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( K `  A ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  A ) )  C_  ( K `  A ) )
311, 24, 30mp2an 654 . . . . . . . . 9  |-  ( ( int `  J ) `
 ( K `  A ) )  C_  ( K `  A )
3229, 31eqsstri 3370 . . . . . . . 8  |-  ( X 
\  ( K `  B ) )  C_  ( K `  A )
332clsss 17110 . . . . . . . 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 1279 . . . . . . 7  |-  ( ( cls `  J ) `
 ( X  \ 
( K `  B
) ) )  C_  ( ( cls `  J
) `  ( K `  A ) )
353fveq1i 5721 . . . . . . 7  |-  ( K `
 ( X  \ 
( K `  B
) ) )  =  ( ( cls `  J
) `  ( X  \  ( K `  B
) ) )
361, 2, 3, 4, 23kur14lem5 24888 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)
373fveq1i 5721 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( ( cls `  J
) `  ( K `  A ) )
3836, 37eqtr3i 2457 . . . . . . 7  |-  ( K `
 A )  =  ( ( cls `  J
) `  ( K `  A ) )
3934, 35, 383sstr4i 3379 . . . . . 6  |-  ( K `
 ( X  \ 
( K `  B
) ) )  C_  ( K `  A )
40 sscon 3473 . . . . . 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 3379 . . . 4  |-  B  C_  ( I `  ( K `  B )
)
432clsss 17110 . . . 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 1279 . . 3  |-  ( ( cls `  J ) `
 B )  C_  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
453fveq1i 5721 . . 3  |-  ( K `
 B )  =  ( ( cls `  J
) `  B )
4644, 45, 153sstr4i 3379 . 2  |-  ( K `
 B )  C_  ( K `  ( I `
 ( K `  B ) ) )
4719, 46eqssi 3356 1  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( K `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    \ cdif 3309    C_ wss 3312   U.cuni 4007   ` cfv 5446   Topctop 16950   intcnt 17073   clsccl 17074
This theorem is referenced by:  kur14lem7  24890
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-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  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-top 16955  df-cld 17075  df-ntr 17076  df-cls 17077
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