Theorem kur14lem6 32571

Description: Lemma for kur14 32576. 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 4059	. . . . . . 7 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ 𝑋
7	5, 6	eqsstri 3949	. . . . . 6 ⊢ 𝐵 ⊆ 𝑋
8	1, 2, 3, 4, 7	kur14lem3 32568	. . . . 5 ⊢ (𝐾‘𝐵) ⊆ 𝑋
9	4	fveq1i 6646	. . . . . 6 ⊢ (𝐼‘(𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐵))
10	2	ntrss2 21662	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵))
11	1, 8, 10	mp2an 691	. . . . . 6 ⊢ ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
12	9, 11	eqsstri 3949	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
13	2	clsss 21659	. . . . 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 1458	. . . 4 ⊢ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐵))
15	3	fveq1i 6646	. . . 4 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
16	3	fveq1i 6646	. . . 4 ⊢ (𝐾‘(𝐾‘𝐵)) = ((cls‘𝐽)‘(𝐾‘𝐵))
17	14, 15, 16	3sstr4i 3958	. . 3 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘(𝐾‘𝐵))
18	1, 2, 3, 4, 7	kur14lem5 32570	. . 3 ⊢ (𝐾‘(𝐾‘𝐵)) = (𝐾‘𝐵)
19	17, 18	sseqtri 3951	. 2 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘𝐵)
20	1, 2, 3, 4, 8	kur14lem2 32567	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
21		difss 4059	. . . . 5 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))) ⊆ 𝑋
22	20, 21	eqsstri 3949	. . . 4 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋
23		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
24	1, 2, 3, 4, 23	kur14lem3 32568	. . . . . . . 8 ⊢ (𝐾‘𝐴) ⊆ 𝑋
25	5	fveq2i 6648	. . . . . . . . . . 11 ⊢ (𝐾‘𝐵) = (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))
26	25	difeq2i 4047	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
27	1, 2, 3, 4, 24	kur14lem2 32567	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
28	4	fveq1i 6646	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴))
29	26, 27, 28	3eqtr2i 2827	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐴))
30	2	ntrss2 21662	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴))
31	1, 24, 30	mp2an 691	. . . . . . . . 9 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴)
32	29, 31	eqsstri 3949	. . . . . . . 8 ⊢ (𝑋 ∖ (𝐾‘𝐵)) ⊆ (𝐾‘𝐴)
33	2	clsss 21659	. . . . . . . 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 1458	. . . . . . 7 ⊢ ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐴))
35	3	fveq1i 6646	. . . . . . 7 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵)))
36	1, 2, 3, 4, 23	kur14lem5 32570	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
37	3	fveq1i 6646	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘(𝐾‘𝐴))
38	36, 37	eqtr3i 2823	. . . . . . 7 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘(𝐾‘𝐴))
39	34, 35, 38	3sstr4i 3958	. . . . . 6 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ (𝐾‘𝐴)
40		sscon 4066	. . . . . 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 3958	. . . 4 ⊢ 𝐵 ⊆ (𝐼‘(𝐾‘𝐵))
43	2	clsss 21659	. . . 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 1458	. . 3 ⊢ ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
45	3	fveq1i 6646	. . 3 ⊢ (𝐾‘𝐵) = ((cls‘𝐽)‘𝐵)
46	44, 45, 15	3sstr4i 3958	. 2 ⊢ (𝐾‘𝐵) ⊆ (𝐾‘(𝐼‘(𝐾‘𝐵)))
47	19, 46	eqssi 3931	1 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)

Syntax hints:   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  intcnt 21622  clsccl 21623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-cld 21624  df-ntr 21625  df-cls 21626

This theorem is referenced by:  kur14lem7  32572