Theorem ntrin 14918

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

Step	Hyp	Ref	Expression
1		inss1 3429	. . . . 5 |- ( A i^i B ) C_ A
2		clscld.1	. . . . . 6 |- X = U. J
3	2	ntrss 14913	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ ( A i^i B ) C_ A ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
4	1, 3	mp3an3 1363	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
5	4	3adant3 1044	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
6		inss2 3430	. . . . 5 |- ( A i^i B ) C_ B
7	2	ntrss 14913	. . . . 5 |- ( ( J e. Top /\ B C_ X /\ ( A i^i B ) C_ B ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
8	6, 7	mp3an3 1363	. . . 4 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
9	8	3adant2 1043	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
10	5, 9	ssind 3433	. 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 1024	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> J e. Top )
12		ssinss1 3438	. . . 4 |- ( A C_ X -> ( A i^i B ) C_ X )
13	12	3ad2ant2 1046	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( A i^i B ) C_ X )
14	2	ntropn 14911	. . . . 5 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) e. J )
15	14	3adant3 1044	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) e. J )
16	2	ntropn 14911	. . . . 5 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
17	16	3adant2 1043	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
18		inopn 14797	. . . 4 |- ( ( J e. Top /\ ( ( int ` J ) ` A ) e. J /\ ( ( int ` J ) ` B ) e. J ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) e. J )
19	11, 15, 17, 18	syl3anc 1274	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) e. J )
20		inss1 3429	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` A )
21	2	ntrss2 14915	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
22	21	3adant3 1044	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) C_ A )
23	20, 22	sstrid 3239	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ A )
24		inss2 3430	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` B )
25	2	ntrss2 14915	. . . . . 6 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
26	25	3adant2 1043	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
27	24, 26	sstrid 3239	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ B )
28	23, 27	ssind 3433	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( A i^i B ) )
29	2	ssntr 14916	. . 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 ) ) )
30	11, 13, 19, 28, 29	syl22anc 1275	. 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 ) ) )
31	10, 30	eqssd 3245	1 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) = ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2202   i^i cin 3200   C_ wss 3201   U.cuni 3898   ` cfv 5333   Topctop 14791   intcnt 14887

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-top 14792  df-ntr 14890

This theorem is referenced by: (None)