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Theorem ntrin 16814
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 3402 . . . . 5  |-  ( A  i^i  B )  C_  A
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32ntrss 16808 . . . . 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 1266 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
543adant3 975 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
6 inss2 3403 . . . . 5  |-  ( A  i^i  B )  C_  B
72ntrss 16808 . . . . 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 1266 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
983adant2 974 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
105, 9ssind 3406 . 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 955 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  J  e.  Top )
12 ssinss1 3410 . . . 4  |-  ( A 
C_  X  ->  ( A  i^i  B )  C_  X )
13123ad2ant2 977 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( A  i^i  B )  C_  X )
142ntropn 16802 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
15143adant3 975 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  e.  J )
162ntropn 16802 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  e.  J )
17163adant2 974 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  e.  J )
18 inopn 16661 . . . 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 1182 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  e.  J )
20 inss1 3402 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  A )
212ntrss2 16810 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
22213adant3 975 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  C_  A )
2320, 22syl5ss 3203 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  A )
24 inss2 3403 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  B )
252ntrss2 16810 . . . . . 6  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  C_  B )
26253adant2 974 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  C_  B )
2724, 26syl5ss 3203 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  B )
2823, 27ssind 3406 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( A  i^i  B ) )
292ssntr 16811 . . 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 1183 . 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 3209 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 set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   U.cuni 3843   ` cfv 5271   Topctop 16647   intcnt 16770
This theorem is referenced by:  dvreslem  19275  dvaddbr  19303  dvmulbr  19304  clsun  26349  neiin  26353
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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