Theorem kur14lem6 31193

Description: Lemma for kur14 31198. If k is the complementation operator and k is the closure operator, this expresses the identity k c k A = k c k c k c k A for any subset A of the topological space. This is the key result that lets us cut down long enough sequences of c k c k ... that arise when applying closure and complement repeatedly to A, and explains why we end up with a number as large as 1 4, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem.j	|- J e. Top
kur14lem.x	|- X = U. J
kur14lem.k	|- K = ( cls ` J )
kur14lem.i	|- I = ( int ` J )
kur14lem.a	|- A C_ X
kur14lem.b	|- B = ( X \ ( K ` A ) )

Assertion

Ref	Expression
kur14lem6	|- ( K ` ( I ` ( K ` B ) ) ) = ( K ` B )

Proof of Theorem kur14lem6

Step	Hyp	Ref	Expression
1		kur14lem.j	. . . . 5 |- J e. Top
2		kur14lem.x	. . . . . 6 |- X = U. J
3		kur14lem.k	. . . . . 6 |- K = ( cls ` J )
4		kur14lem.i	. . . . . 6 |- I = ( int ` J )
5		kur14lem.b	. . . . . . 7 |- B = ( X \ ( K ` A ) )
6		difss 3737	. . . . . . 7 |- ( X \ ( K ` A ) ) C_ X
7	5, 6	eqsstri 3635	. . . . . 6 |- B C_ X
8	1, 2, 3, 4, 7	kur14lem3 31190	. . . . 5 |- ( K ` B ) C_ X
9	4	fveq1i 6192	. . . . . 6 |- ( I ` ( K ` B ) ) = ( ( int ` J ) ` ( K ` B ) )
10	2	ntrss2 20861	. . . . . . 7 |- ( ( J e. Top /\ ( K ` B ) C_ X ) -> ( ( int ` J ) ` ( K ` B ) ) C_ ( K ` B ) )
11	1, 8, 10	mp2an 708	. . . . . 6 |- ( ( int ` J ) ` ( K ` B ) ) C_ ( K ` B )
12	9, 11	eqsstri 3635	. . . . 5 |- ( I ` ( K ` B ) ) C_ ( K ` B )
13	2	clsss 20858	. . . . 5 |- ( ( J e. Top /\ ( K ` B ) C_ X /\ ( I ` ( K ` B ) ) C_ ( K ` B ) ) -> ( ( cls ` J ) ` ( I ` ( K ` B ) ) ) C_ ( ( cls ` J ) ` ( K ` B ) ) )
14	1, 8, 12, 13	mp3an 1424	. . . 4 |- ( ( cls ` J ) ` ( I ` ( K ` B ) ) ) C_ ( ( cls ` J ) ` ( K ` B ) )
15	3	fveq1i 6192	. . . 4 |- ( K ` ( I ` ( K ` B ) ) ) = ( ( cls ` J ) ` ( I ` ( K ` B ) ) )
16	3	fveq1i 6192	. . . 4 |- ( K ` ( K ` B ) ) = ( ( cls ` J ) ` ( K ` B ) )
17	14, 15, 16	3sstr4i 3644	. . 3 |- ( K ` ( I ` ( K ` B ) ) ) C_ ( K ` ( K ` B ) )
18	1, 2, 3, 4, 7	kur14lem5 31192	. . 3 |- ( K ` ( K ` B ) ) = ( K ` B )
19	17, 18	sseqtri 3637	. 2 |- ( K ` ( I ` ( K ` B ) ) ) C_ ( K ` B )
20	1, 2, 3, 4, 8	kur14lem2 31189	. . . . 5 |- ( I ` ( K ` B ) ) = ( X \ ( K ` ( X \ ( K ` B ) ) ) )
21		difss 3737	. . . . 5 |- ( X \ ( K ` ( X \ ( K ` B ) ) ) ) C_ X
22	20, 21	eqsstri 3635	. . . 4 |- ( I ` ( K ` B ) ) C_ X
23		kur14lem.a	. . . . . . . . 9 |- A C_ X
24	1, 2, 3, 4, 23	kur14lem3 31190	. . . . . . . 8 |- ( K ` A ) C_ X
25	5	fveq2i 6194	. . . . . . . . . . 11 |- ( K ` B ) = ( K ` ( X \ ( K ` A ) ) )
26	25	difeq2i 3725	. . . . . . . . . 10 |- ( X \ ( K ` B ) ) = ( X \ ( K ` ( X \ ( K ` A ) ) ) )
27	1, 2, 3, 4, 24	kur14lem2 31189	. . . . . . . . . 10 |- ( I ` ( K ` A ) ) = ( X \ ( K ` ( X \ ( K ` A ) ) ) )
28	4	fveq1i 6192	. . . . . . . . . 10 |- ( I ` ( K ` A ) ) = ( ( int ` J ) ` ( K ` A ) )
29	26, 27, 28	3eqtr2i 2650	. . . . . . . . 9 |- ( X \ ( K ` B ) ) = ( ( int ` J ) ` ( K ` A ) )
30	2	ntrss2 20861	. . . . . . . . . 10 |- ( ( J e. Top /\ ( K ` A ) C_ X ) -> ( ( int ` J ) ` ( K ` A ) ) C_ ( K ` A ) )
31	1, 24, 30	mp2an 708	. . . . . . . . 9 |- ( ( int ` J ) ` ( K ` A ) ) C_ ( K ` A )
32	29, 31	eqsstri 3635	. . . . . . . 8 |- ( X \ ( K ` B ) ) C_ ( K ` A )
33	2	clsss 20858	. . . . . . . 8 |- ( ( J e. Top /\ ( K ` A ) C_ X /\ ( X \ ( K ` B ) ) C_ ( K ` A ) ) -> ( ( cls ` J ) ` ( X \ ( K ` B ) ) ) C_ ( ( cls ` J ) ` ( K ` A ) ) )
34	1, 24, 32, 33	mp3an 1424	. . . . . . 7 |- ( ( cls ` J ) ` ( X \ ( K ` B ) ) ) C_ ( ( cls ` J ) ` ( K ` A ) )
35	3	fveq1i 6192	. . . . . . 7 |- ( K ` ( X \ ( K ` B ) ) ) = ( ( cls ` J ) ` ( X \ ( K ` B ) ) )
36	1, 2, 3, 4, 23	kur14lem5 31192	. . . . . . . 8 |- ( K ` ( K ` A ) ) = ( K ` A )
37	3	fveq1i 6192	. . . . . . . 8 |- ( K ` ( K ` A ) ) = ( ( cls ` J ) ` ( K ` A ) )
38	36, 37	eqtr3i 2646	. . . . . . 7 |- ( K ` A ) = ( ( cls ` J ) ` ( K ` A ) )
39	34, 35, 38	3sstr4i 3644	. . . . . 6 |- ( K ` ( X \ ( K ` B ) ) ) C_ ( K ` A )
40		sscon 3744	. . . . . 6 |- ( ( K ` ( X \ ( K ` B ) ) ) C_ ( K ` A ) -> ( X \ ( K ` A ) ) C_ ( X \ ( K ` ( X \ ( K ` B ) ) ) ) )
41	39, 40	ax-mp 5	. . . . 5 |- ( X \ ( K ` A ) ) C_ ( X \ ( K ` ( X \ ( K ` B ) ) ) )
42	41, 5, 20	3sstr4i 3644	. . . 4 |- B C_ ( I ` ( K ` B ) )
43	2	clsss 20858	. . . 4 |- ( ( J e. Top /\ ( I ` ( K ` B ) ) C_ X /\ B C_ ( I ` ( K ` B ) ) ) -> ( ( cls ` J ) ` B ) C_ ( ( cls ` J ) ` ( I ` ( K ` B ) ) ) )
44	1, 22, 42, 43	mp3an 1424	. . 3 |- ( ( cls ` J ) ` B ) C_ ( ( cls ` J ) ` ( I ` ( K ` B ) ) )
45	3	fveq1i 6192	. . 3 |- ( K ` B ) = ( ( cls ` J ) ` B )
46	44, 45, 15	3sstr4i 3644	. 2 |- ( K ` B ) C_ ( K ` ( I ` ( K ` B ) ) )
47	19, 46	eqssi 3619	1 |- ( K ` ( I ` ( K ` B ) ) ) = ( K ` B )

Syntax hints:   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   intcnt 20821   clsccl 20822

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:  kur14lem7  31194