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Theorem kur14lem6 27121
Description: Lemma for kur14 27126. 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 3504 . . . . . . 7  |-  ( X 
\  ( K `  A ) )  C_  X
75, 6eqsstri 3407 . . . . . 6  |-  B  C_  X
81, 2, 3, 4, 7kur14lem3 27118 . . . . 5  |-  ( K `
 B )  C_  X
94fveq1i 5713 . . . . . 6  |-  ( I `
 ( K `  B ) )  =  ( ( int `  J
) `  ( K `  B ) )
102ntrss2 18683 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( K `  B ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  B ) )  C_  ( K `  B ) )
111, 8, 10mp2an 672 . . . . . 6  |-  ( ( int `  J ) `
 ( K `  B ) )  C_  ( K `  B )
129, 11eqsstri 3407 . . . . 5  |-  ( I `
 ( K `  B ) )  C_  ( K `  B )
132clsss 18680 . . . . 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 1314 . . . 4  |-  ( ( cls `  J ) `
 ( I `  ( K `  B ) ) )  C_  (
( cls `  J
) `  ( K `  B ) )
153fveq1i 5713 . . . 4  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
163fveq1i 5713 . . . 4  |-  ( K `
 ( K `  B ) )  =  ( ( cls `  J
) `  ( K `  B ) )
1714, 15, 163sstr4i 3416 . . 3  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  ( K `  B ) )
181, 2, 3, 4, 7kur14lem5 27120 . . 3  |-  ( K `
 ( K `  B ) )  =  ( K `  B
)
1917, 18sseqtri 3409 . 2  |-  ( K `
 ( I `  ( K `  B ) ) )  C_  ( K `  B )
201, 2, 3, 4, 8kur14lem2 27117 . . . . 5  |-  ( I `
 ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  B
) ) ) )
21 difss 3504 . . . . 5  |-  ( X 
\  ( K `  ( X  \  ( K `  B )
) ) )  C_  X
2220, 21eqsstri 3407 . . . 4  |-  ( I `
 ( K `  B ) )  C_  X
23 kur14lem.a . . . . . . . . 9  |-  A  C_  X
241, 2, 3, 4, 23kur14lem3 27118 . . . . . . . 8  |-  ( K `
 A )  C_  X
255fveq2i 5715 . . . . . . . . . . 11  |-  ( K `
 B )  =  ( K `  ( X  \  ( K `  A ) ) )
2625difeq2i 3492 . . . . . . . . . 10  |-  ( X 
\  ( K `  B ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
271, 2, 3, 4, 24kur14lem2 27117 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( X  \  ( K `  ( X  \  ( K `  A
) ) ) )
284fveq1i 5713 . . . . . . . . . 10  |-  ( I `
 ( K `  A ) )  =  ( ( int `  J
) `  ( K `  A ) )
2926, 27, 283eqtr2i 2469 . . . . . . . . 9  |-  ( X 
\  ( K `  B ) )  =  ( ( int `  J
) `  ( K `  A ) )
302ntrss2 18683 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( K `  A ) 
C_  X )  -> 
( ( int `  J
) `  ( K `  A ) )  C_  ( K `  A ) )
311, 24, 30mp2an 672 . . . . . . . . 9  |-  ( ( int `  J ) `
 ( K `  A ) )  C_  ( K `  A )
3229, 31eqsstri 3407 . . . . . . . 8  |-  ( X 
\  ( K `  B ) )  C_  ( K `  A )
332clsss 18680 . . . . . . . 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 1314 . . . . . . 7  |-  ( ( cls `  J ) `
 ( X  \ 
( K `  B
) ) )  C_  ( ( cls `  J
) `  ( K `  A ) )
353fveq1i 5713 . . . . . . 7  |-  ( K `
 ( X  \ 
( K `  B
) ) )  =  ( ( cls `  J
) `  ( X  \  ( K `  B
) ) )
361, 2, 3, 4, 23kur14lem5 27120 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( K `  A
)
373fveq1i 5713 . . . . . . . 8  |-  ( K `
 ( K `  A ) )  =  ( ( cls `  J
) `  ( K `  A ) )
3836, 37eqtr3i 2465 . . . . . . 7  |-  ( K `
 A )  =  ( ( cls `  J
) `  ( K `  A ) )
3934, 35, 383sstr4i 3416 . . . . . 6  |-  ( K `
 ( X  \ 
( K `  B
) ) )  C_  ( K `  A )
40 sscon 3511 . . . . . 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 3416 . . . 4  |-  B  C_  ( I `  ( K `  B )
)
432clsss 18680 . . . 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 1314 . . 3  |-  ( ( cls `  J ) `
 B )  C_  ( ( cls `  J
) `  ( I `  ( K `  B
) ) )
453fveq1i 5713 . . 3  |-  ( K `
 B )  =  ( ( cls `  J
) `  B )
4644, 45, 153sstr4i 3416 . 2  |-  ( K `
 B )  C_  ( K `  ( I `
 ( K `  B ) ) )
4719, 46eqssi 3393 1  |-  ( K `
 ( I `  ( K `  B ) ) )  =  ( K `  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756    \ cdif 3346    C_ wss 3349   U.cuni 4112   ` cfv 5439   Topctop 18520   intcnt 18643   clsccl 18644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-top 18525  df-cld 18645  df-ntr 18646  df-cls 18647
This theorem is referenced by:  kur14lem7  27122
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