Theorem ntrin 20865

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 3833	. . . . 5 |- ( A i^i B ) C_ A
2		clscld.1	. . . . . 6 |- X = U. J
3	2	ntrss 20859	. . . . 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 1413	. . . 4 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
5	4	3adant3 1081	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` A ) )
6		inss2 3834	. . . . 5 |- ( A i^i B ) C_ B
7	2	ntrss 20859	. . . . 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 1413	. . . 4 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
9	8	3adant2 1080	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` ( A i^i B ) ) C_ ( ( int ` J ) ` B ) )
10	5, 9	ssind 3837	. 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 1061	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> J e. Top )
12		ssinss1 3841	. . . 4 |- ( A C_ X -> ( A i^i B ) C_ X )
13	12	3ad2ant2 1083	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( A i^i B ) C_ X )
14	2	ntropn 20853	. . . . 5 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) e. J )
15	14	3adant3 1081	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) e. J )
16	2	ntropn 20853	. . . . 5 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
17	16	3adant2 1080	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) e. J )
18		inopn 20704	. . . 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 1326	. . 3 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) e. J )
20		inss1 3833	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` A )
21	2	ntrss2 20861	. . . . . 6 |- ( ( J e. Top /\ A C_ X ) -> ( ( int ` J ) ` A ) C_ A )
22	21	3adant3 1081	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` A ) C_ A )
23	20, 22	syl5ss 3614	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ A )
24		inss2 3834	. . . . 5 |- ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ ( ( int ` J ) ` B )
25	2	ntrss2 20861	. . . . . 6 |- ( ( J e. Top /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
26	25	3adant2 1080	. . . . 5 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( int ` J ) ` B ) C_ B )
27	24, 26	syl5ss 3614	. . . 4 |- ( ( J e. Top /\ A C_ X /\ B C_ X ) -> ( ( ( int ` J ) ` A ) i^i ( ( int ` J ) ` B ) ) C_ B )
28	23, 27	ssind 3837	. . 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 20862	. . 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 1327	. 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 3620	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 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   intcnt 20821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  dvreslem  23673  dvaddbr  23701  dvmulbr  23702  clsun  32323  neiin  32327