Theorem kur14lem6 35745

Description: Lemma for kur14 35750. 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 4090	. . . . . . 7 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ 𝑋
7	5, 6	eqsstri 3984	. . . . . 6 ⊢ 𝐵 ⊆ 𝑋
8	1, 2, 3, 4, 7	kur14lem3 35742	. . . . 5 ⊢ (𝐾‘𝐵) ⊆ 𝑋
9	4	fveq1i 6887	. . . . . 6 ⊢ (𝐼‘(𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐵))
10	2	ntrss2 23269	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵))
11	1, 8, 10	mp2an 705	. . . . . 6 ⊢ ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
12	9, 11	eqsstri 3984	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
13	2	clsss 23266	. . . . 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 1490	. . . 4 ⊢ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐵))
15	3	fveq1i 6887	. . . 4 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
16	3	fveq1i 6887	. . . 4 ⊢ (𝐾‘(𝐾‘𝐵)) = ((cls‘𝐽)‘(𝐾‘𝐵))
17	14, 15, 16	3sstr4i 3989	. . 3 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘(𝐾‘𝐵))
18	1, 2, 3, 4, 7	kur14lem5 35744	. . 3 ⊢ (𝐾‘(𝐾‘𝐵)) = (𝐾‘𝐵)
19	17, 18	sseqtri 3986	. 2 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘𝐵)
20	1, 2, 3, 4, 8	kur14lem2 35741	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
21		difss 4090	. . . . 5 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))) ⊆ 𝑋
22	20, 21	eqsstri 3984	. . . 4 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋
23		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
24	1, 2, 3, 4, 23	kur14lem3 35742	. . . . . . . 8 ⊢ (𝐾‘𝐴) ⊆ 𝑋
25	5	fveq2i 6889	. . . . . . . . . . 11 ⊢ (𝐾‘𝐵) = (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))
26	25	difeq2i 4078	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
27	1, 2, 3, 4, 24	kur14lem2 35741	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
28	4	fveq1i 6887	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴))
29	26, 27, 28	3eqtr2i 2794	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐴))
30	2	ntrss2 23269	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴))
31	1, 24, 30	mp2an 705	. . . . . . . . 9 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴)
32	29, 31	eqsstri 3984	. . . . . . . 8 ⊢ (𝑋 ∖ (𝐾‘𝐵)) ⊆ (𝐾‘𝐴)
33	2	clsss 23266	. . . . . . . 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 1490	. . . . . . 7 ⊢ ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐴))
35	3	fveq1i 6887	. . . . . . 7 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵)))
36	1, 2, 3, 4, 23	kur14lem5 35744	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
37	3	fveq1i 6887	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘(𝐾‘𝐴))
38	36, 37	eqtr3i 2790	. . . . . . 7 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘(𝐾‘𝐴))
39	34, 35, 38	3sstr4i 3989	. . . . . 6 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ (𝐾‘𝐴)
40		sscon 4097	. . . . . 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 3989	. . . 4 ⊢ 𝐵 ⊆ (𝐼‘(𝐾‘𝐵))
43	2	clsss 23266	. . . 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 1490	. . 3 ⊢ ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
45	3	fveq1i 6887	. . 3 ⊢ (𝐾‘𝐵) = ((cls‘𝐽)‘𝐵)
46	44, 45, 15	3sstr4i 3989	. 2 ⊢ (𝐾‘𝐵) ⊆ (𝐾‘(𝐼‘(𝐾‘𝐵)))
47	19, 46	eqssi 3954	1 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4874  ‘cfv 6541  Topctop 23105  intcnt 23229  clsccl 23230

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7743

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-top 23106  df-cld 23231  df-ntr 23232  df-cls 23233

This theorem is used by:  kur14lem7  35746