Theorem ntrin 14360

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 3383	. . . . 5 |- ( A i^i B ) C_ A
2		clscld.1	. . . . . 6 |- X = U. J
3	2	ntrss 14355	. . . . 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 1337	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
5	4	3adant3 1019	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
6		inss2 3384	. . . . 5 |- ( A i^i B ) C_ B
7	2	ntrss 14355	. . . . 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 1337	. . . 4 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
9	8	3adant2 1018	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
10	5, 9	ssind 3387	. 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 999	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> J e. Top )
12		ssinss1 3392	. . . 4 |- ( A C_ X -> ( A i^i B ) C_ X )
13	12	3ad2ant2 1021	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( A i^i B ) C_ X )
14	2	ntropn 14353	. . . . 5 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) e. J )
15	14	3adant3 1019	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) e. J )
16	2	ntropn 14353	. . . . 5 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
17	16	3adant2 1018	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
18		inopn 14239	. . . 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 1249	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) e. J )
20		inss1 3383	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` A )
21	2	ntrss2 14357	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
22	21	3adant3 1019	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) C_ A )
23	20, 22	sstrid 3194	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ A )
24		inss2 3384	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` B )
25	2	ntrss2 14357	. . . . . 6 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
26	25	3adant2 1018	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
27	24, 26	sstrid 3194	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ B )
28	23, 27	ssind 3387	. . 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 14358	. . 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 1250	. 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 3200	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 980   = wceq 1364   e. wcel 2167   i^i cin 3156   C_ wss 3157   U.cuni 3839   ` cfv 5258   Topctop 14233   intcnt 14329

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-top 14234  df-ntr 14332

This theorem is referenced by: (None)