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Theorem ntrin 11991
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 3235 . . . . 5  |-  ( A  i^i  B )  C_  A
2 clscld.1 . . . . . 6  |-  X  = 
U. J
32ntrss 11986 . . . . 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 1269 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
543adant3 966 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  A )
)
6 inss2 3236 . . . . 5  |-  ( A  i^i  B )  C_  B
72ntrss 11986 . . . . 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 1269 . . . 4  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
983adant2 965 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  ( A  i^i  B ) )  C_  ( ( int `  J
) `  B )
)
105, 9ssind 3239 . 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 946 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  J  e.  Top )
12 ssinss1 3244 . . . 4  |-  ( A 
C_  X  ->  ( A  i^i  B )  C_  X )
13123ad2ant2 968 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  ( A  i^i  B )  C_  X )
142ntropn 11984 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  e.  J )
15143adant3 966 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  e.  J )
162ntropn 11984 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  e.  J )
17163adant2 965 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  e.  J )
18 inopn 11869 . . . 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 1181 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  e.  J )
20 inss1 3235 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  A )
212ntrss2 11988 . . . . . 6  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( int `  J
) `  A )  C_  A )
22213adant3 966 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  A )  C_  A )
2320, 22syl5ss 3050 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  A )
24 inss2 3236 . . . . 5  |-  ( ( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( ( int `  J ) `  B )
252ntrss2 11988 . . . . . 6  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( ( int `  J
) `  B )  C_  B )
26253adant2 965 . . . . 5  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( int `  J
) `  B )  C_  B )
2724, 26syl5ss 3050 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  B )
2823, 27ssind 3239 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  X )  ->  (
( ( int `  J
) `  A )  i^i  ( ( int `  J
) `  B )
)  C_  ( A  i^i  B ) )
292ssntr 11989 . . 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 1182 . 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 3056 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 927    = wceq 1296    e. wcel 1445    i^i cin 3012    C_ wss 3013   U.cuni 3675   ` cfv 5049   Topctop 11863   intcnt 11960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-top 11864  df-ntr 11963
This theorem is referenced by: (None)
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