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Theorem ntrin 20153
Description: A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrin  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  =  ( ( ( int `  J ) `  A
)  i^i  ( ( int `  J ) `  B ) ) )

Proof of Theorem ntrin
StepHypRef Expression
1 inss1 3643 . . . . 5  |-  ( A  i^i  B )  C_  A
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32ntrss 20147 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  ( A  i^i  B )  C_  A )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
41, 3mp3an3 1379 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
543adant3 1050 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
6 inss2 3644 . . . . 5  |-  ( A  i^i  B )  C_  B
72ntrss 20147 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X  /\  ( A  i^i  B )  C_  B )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
86, 7mp3an3 1379 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
983adant2 1049 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
105, 9ssind 3647 . 2  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( ( int `  J ) `  A
)  i^i  ( ( int `  J ) `  B ) ) )
11 simp1 1030 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  J  e.  Top )
12 ssinss1 3651 . . . 4  |-  ( A 
C_  X  ->  ( A  i^i  B )  C_  X )
13123ad2ant2 1052 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( A  i^i  B )  C_  X )
142ntropn 20141 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
15143adant3 1050 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  e.  J )
162ntropn 20141 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  e.  J )
17163adant2 1049 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  e.  J )
18 inopn 20006 . . . 4  |-  ( ( J  e.  Top  /\  ( ( int `  J
) `  A )  e.  J  /\  (
( int `  J
) `  B )  e.  J )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  e.  J )
1911, 15, 17, 18syl3anc 1292 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  e.  J )
20 inss1 3643 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  A )
212ntrss2 20149 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
22213adant3 1050 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  C_  A )
2320, 22syl5ss 3429 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  A )
24 inss2 3644 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  B )
252ntrss2 20149 . . . . . 6  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  C_  B )
26253adant2 1049 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  C_  B )
2724, 26syl5ss 3429 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  B )
2823, 27ssind 3647 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( A  i^i  B ) )
292ssntr 20150 . . 3  |-  ( ( ( J  e.  Top  /\  ( A  i^i  B
)  C_  X )  /\  ( ( ( ( int `  J ) `
 A )  i^i  ( ( int `  J
) `  B )
)  e.  J  /\  ( ( ( int `  J ) `  A
)  i^i  ( ( int `  J ) `  B ) )  C_  ( A  i^i  B ) ) )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  ( A  i^i  B ) ) )
3011, 13, 19, 28, 29syl22anc 1293 . 2  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  ( A  i^i  B ) ) )
3110, 30eqssd 3435 1  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  =  ( ( ( int `  J ) `  A
)  i^i  ( ( int `  J ) `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   U.cuni 4190   ` cfv 5589   Topctop 19994   intcnt 20109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-top 19998  df-cld 20111  df-ntr 20112  df-cls 20113
This theorem is referenced by:  dvreslem  22943  dvaddbr  22971  dvmulbr  22972  clsun  31055  neiin  31059
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