Theorem kur14lem6 34952

Description: Lemma for kur14 34957. If 𝑘 is the complementation operator and 𝑘 is the closure operator, this expresses the identity 𝑘𝑐𝑘𝐴 = 𝑘𝑐𝑘𝑐𝑘𝑐𝑘𝐴 for any subset 𝐴 of the topological space. This is the key result that lets us cut down long enough sequences of 𝑐𝑘𝑐𝑘... that arise when applying closure and complement repeatedly to 𝐴, and explains why we end up with a number as large as 14, yet no larger. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem.j	⊢ 𝐽 ∈ Top
kur14lem.x	⊢ 𝑋 = ∪ 𝐽
kur14lem.k	⊢ 𝐾 = (cls‘𝐽)
kur14lem.i	⊢ 𝐼 = (int‘𝐽)
kur14lem.a	⊢ 𝐴 ⊆ 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 ∈ Top
2		kur14lem.x	. . . . . 6 ⊢ 𝑋 = ∪ 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 4128	. . . . . . 7 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ 𝑋
7	5, 6	eqsstri 4011	. . . . . 6 ⊢ 𝐵 ⊆ 𝑋
8	1, 2, 3, 4, 7	kur14lem3 34949	. . . . 5 ⊢ (𝐾‘𝐵) ⊆ 𝑋
9	4	fveq1i 6897	. . . . . 6 ⊢ (𝐼‘(𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐵))
10	2	ntrss2 23005	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵))
11	1, 8, 10	mp2an 690	. . . . . 6 ⊢ ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
12	9, 11	eqsstri 4011	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
13	2	clsss 23002	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐵) ⊆ 𝑋 ∧ (𝐼‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)) → ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐵)))
14	1, 8, 12, 13	mp3an 1457	. . . 4 ⊢ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐵))
15	3	fveq1i 6897	. . . 4 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
16	3	fveq1i 6897	. . . 4 ⊢ (𝐾‘(𝐾‘𝐵)) = ((cls‘𝐽)‘(𝐾‘𝐵))
17	14, 15, 16	3sstr4i 4020	. . 3 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘(𝐾‘𝐵))
18	1, 2, 3, 4, 7	kur14lem5 34951	. . 3 ⊢ (𝐾‘(𝐾‘𝐵)) = (𝐾‘𝐵)
19	17, 18	sseqtri 4013	. 2 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘𝐵)
20	1, 2, 3, 4, 8	kur14lem2 34948	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
21		difss 4128	. . . . 5 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))) ⊆ 𝑋
22	20, 21	eqsstri 4011	. . . 4 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋
23		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
24	1, 2, 3, 4, 23	kur14lem3 34949	. . . . . . . 8 ⊢ (𝐾‘𝐴) ⊆ 𝑋
25	5	fveq2i 6899	. . . . . . . . . . 11 ⊢ (𝐾‘𝐵) = (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))
26	25	difeq2i 4115	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
27	1, 2, 3, 4, 24	kur14lem2 34948	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
28	4	fveq1i 6897	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴))
29	26, 27, 28	3eqtr2i 2759	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐴))
30	2	ntrss2 23005	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴))
31	1, 24, 30	mp2an 690	. . . . . . . . 9 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴)
32	29, 31	eqsstri 4011	. . . . . . . 8 ⊢ (𝑋 ∖ (𝐾‘𝐵)) ⊆ (𝐾‘𝐴)
33	2	clsss 23002	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐴) ⊆ 𝑋 ∧ (𝑋 ∖ (𝐾‘𝐵)) ⊆ (𝐾‘𝐴)) → ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐴)))
34	1, 24, 32, 33	mp3an 1457	. . . . . . 7 ⊢ ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐴))
35	3	fveq1i 6897	. . . . . . 7 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵)))
36	1, 2, 3, 4, 23	kur14lem5 34951	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
37	3	fveq1i 6897	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘(𝐾‘𝐴))
38	36, 37	eqtr3i 2755	. . . . . . 7 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘(𝐾‘𝐴))
39	34, 35, 38	3sstr4i 4020	. . . . . 6 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ (𝐾‘𝐴)
40		sscon 4135	. . . . . 6 ⊢ ((𝐾‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ (𝐾‘𝐴) → (𝑋 ∖ (𝐾‘𝐴)) ⊆ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))))
41	39, 40	ax-mp 5	. . . . 5 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
42	41, 5, 20	3sstr4i 4020	. . . 4 ⊢ 𝐵 ⊆ (𝐼‘(𝐾‘𝐵))
43	2	clsss 23002	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋 ∧ 𝐵 ⊆ (𝐼‘(𝐾‘𝐵))) → ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))))
44	1, 22, 42, 43	mp3an 1457	. . 3 ⊢ ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
45	3	fveq1i 6897	. . 3 ⊢ (𝐾‘𝐵) = ((cls‘𝐽)‘𝐵)
46	44, 45, 15	3sstr4i 4020	. 2 ⊢ (𝐾‘𝐵) ⊆ (𝐾‘(𝐼‘(𝐾‘𝐵)))
47	19, 46	eqssi 3993	1 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)

Syntax hints:   = wceq 1533   ∈ wcel 2098   ∖ cdif 3941   ⊆ wss 3944  ∪ cuni 4909  ‘cfv 6549  Topctop 22839  intcnt 22965  clsccl 22966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-top 22840  df-cld 22967  df-ntr 22968  df-cls 22969

This theorem is referenced by:  kur14lem7  34953