ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ntrin Unicode version

Theorem ntrin 13288
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 3355 . . . . 5  |-  ( A  i^i  B )  C_  A
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32ntrss 13283 . . . . 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 1326 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
543adant3 1017 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
6 inss2 3356 . . . . 5  |-  ( A  i^i  B )  C_  B
72ntrss 13283 . . . . 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 1326 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
983adant2 1016 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
105, 9ssind 3359 . 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 997 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  J  e.  Top )
12 ssinss1 3364 . . . 4  |-  ( A 
C_  X  ->  ( A  i^i  B )  C_  X )
13123ad2ant2 1019 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( A  i^i  B )  C_  X )
142ntropn 13281 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
15143adant3 1017 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  e.  J )
162ntropn 13281 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  e.  J )
17163adant2 1016 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  e.  J )
18 inopn 13165 . . . 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 1238 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  e.  J )
20 inss1 3355 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  A )
212ntrss2 13285 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
22213adant3 1017 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  C_  A )
2320, 22sstrid 3166 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  A )
24 inss2 3356 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  B )
252ntrss2 13285 . . . . . 6  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  C_  B )
26253adant2 1016 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  C_  B )
2724, 26sstrid 3166 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  B )
2823, 27ssind 3359 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( A  i^i  B ) )
292ssntr 13286 . . 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 1239 . 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 3172 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 978    = wceq 1353    e. wcel 2148    i^i cin 3128    C_ wss 3129   U.cuni 3807   ` cfv 5212   Topctop 13159   intcnt 13257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-top 13160  df-ntr 13260
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator