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

Theorem ntrin 12302
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 3296 . . . . 5  |-  ( A  i^i  B )  C_  A
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32ntrss 12297 . . . . 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 1304 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
543adant3 1001 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
6 inss2 3297 . . . . 5  |-  ( A  i^i  B )  C_  B
72ntrss 12297 . . . . 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 1304 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
983adant2 1000 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
105, 9ssind 3300 . 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 981 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  J  e.  Top )
12 ssinss1 3305 . . . 4  |-  ( A 
C_  X  ->  ( A  i^i  B )  C_  X )
13123ad2ant2 1003 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( A  i^i  B )  C_  X )
142ntropn 12295 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
15143adant3 1001 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  e.  J )
162ntropn 12295 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  e.  J )
17163adant2 1000 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  e.  J )
18 inopn 12179 . . . 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 1216 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  e.  J )
20 inss1 3296 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  A )
212ntrss2 12299 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
22213adant3 1001 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  C_  A )
2320, 22sstrid 3108 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  A )
24 inss2 3297 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  B )
252ntrss2 12299 . . . . . 6  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  C_  B )
26253adant2 1000 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  C_  B )
2724, 26sstrid 3108 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  B )
2823, 27ssind 3300 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( A  i^i  B ) )
292ssntr 12300 . . 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 1217 . 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 3114 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 962    = wceq 1331    e. wcel 1480    i^i cin 3070    C_ wss 3071   U.cuni 3736   ` cfv 5123   Topctop 12173   intcnt 12271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12174  df-ntr 12274
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator