Theorem ntrin 12764

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 3342	. . . . 5 |- ( A i^i B ) C_ A
2		clscld.1	. . . . . 6 |- X = U. J
3	2	ntrss 12759	. . . . 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 1316	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
5	4	3adant3 1007	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
6		inss2 3343	. . . . 5 |- ( A i^i B ) C_ B
7	2	ntrss 12759	. . . . 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 1316	. . . 4 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
9	8	3adant2 1006	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
10	5, 9	ssind 3346	. 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 987	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> J e. Top )
12		ssinss1 3351	. . . 4 |- ( A C_ X -> ( A i^i B ) C_ X )
13	12	3ad2ant2 1009	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( A i^i B ) C_ X )
14	2	ntropn 12757	. . . . 5 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) e. J )
15	14	3adant3 1007	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) e. J )
16	2	ntropn 12757	. . . . 5 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
17	16	3adant2 1006	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
18		inopn 12641	. . . 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 1228	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) e. J )
20		inss1 3342	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` A )
21	2	ntrss2 12761	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
22	21	3adant3 1007	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) C_ A )
23	20, 22	sstrid 3153	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ A )
24		inss2 3343	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` B )
25	2	ntrss2 12761	. . . . . 6 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
26	25	3adant2 1006	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
27	24, 26	sstrid 3153	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ B )
28	23, 27	ssind 3346	. . 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 12762	. . 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 1229	. 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 3159	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 968   = wceq 1343   e. wcel 2136   i^i cin 3115   C_ wss 3116   U.cuni 3789   ` cfv 5188   Topctop 12635   intcnt 12733

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-ntr 12736

This theorem is referenced by: (None)