Theorem kur14lem6 35171

Description: Lemma for kur14 35176. 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 4153	. . . . . . 7 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ 𝑋
7	5, 6	eqsstri 4037	. . . . . 6 ⊢ 𝐵 ⊆ 𝑋
8	1, 2, 3, 4, 7	kur14lem3 35168	. . . . 5 ⊢ (𝐾‘𝐵) ⊆ 𝑋
9	4	fveq1i 6920	. . . . . 6 ⊢ (𝐼‘(𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐵))
10	2	ntrss2 23079	. . . . . . 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 4037	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
13	2	clsss 23076	. . . . 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 1461	. . . 4 ⊢ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐵))
15	3	fveq1i 6920	. . . 4 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
16	3	fveq1i 6920	. . . 4 ⊢ (𝐾‘(𝐾‘𝐵)) = ((cls‘𝐽)‘(𝐾‘𝐵))
17	14, 15, 16	3sstr4i 4046	. . 3 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘(𝐾‘𝐵))
18	1, 2, 3, 4, 7	kur14lem5 35170	. . 3 ⊢ (𝐾‘(𝐾‘𝐵)) = (𝐾‘𝐵)
19	17, 18	sseqtri 4039	. 2 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘𝐵)
20	1, 2, 3, 4, 8	kur14lem2 35167	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
21		difss 4153	. . . . 5 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))) ⊆ 𝑋
22	20, 21	eqsstri 4037	. . . 4 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋
23		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
24	1, 2, 3, 4, 23	kur14lem3 35168	. . . . . . . 8 ⊢ (𝐾‘𝐴) ⊆ 𝑋
25	5	fveq2i 6922	. . . . . . . . . . 11 ⊢ (𝐾‘𝐵) = (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))
26	25	difeq2i 4140	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
27	1, 2, 3, 4, 24	kur14lem2 35167	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
28	4	fveq1i 6920	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = ((int‘𝐽)‘(𝐾‘𝐴))
29	26, 27, 28	3eqtr2i 2768	. . . . . . . . 9 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐴))
30	2	ntrss2 23079	. . . . . . . . . 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 4037	. . . . . . . 8 ⊢ (𝑋 ∖ (𝐾‘𝐵)) ⊆ (𝐾‘𝐴)
33	2	clsss 23076	. . . . . . . 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 1461	. . . . . . 7 ⊢ ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐴))
35	3	fveq1i 6920	. . . . . . 7 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵)))
36	1, 2, 3, 4, 23	kur14lem5 35170	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
37	3	fveq1i 6920	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘(𝐾‘𝐴))
38	36, 37	eqtr3i 2764	. . . . . . 7 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘(𝐾‘𝐴))
39	34, 35, 38	3sstr4i 4046	. . . . . 6 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ (𝐾‘𝐴)
40		sscon 4160	. . . . . 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 4046	. . . 4 ⊢ 𝐵 ⊆ (𝐼‘(𝐾‘𝐵))
43	2	clsss 23076	. . . 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 1461	. . 3 ⊢ ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
45	3	fveq1i 6920	. . 3 ⊢ (𝐾‘𝐵) = ((cls‘𝐽)‘𝐵)
46	44, 45, 15	3sstr4i 4046	. 2 ⊢ (𝐾‘𝐵) ⊆ (𝐾‘(𝐼‘(𝐾‘𝐵)))
47	19, 46	eqssi 4019	1 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)

Syntax hints:   = wceq 1537   ∈ wcel 2103   ∖ cdif 3967   ⊆ wss 3970  ∪ cuni 4931  ‘cfv 6572  Topctop 22913  intcnt 23039  clsccl 23040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-top 22914  df-cld 23041  df-ntr 23042  df-cls 23043

This theorem is referenced by:  kur14lem7  35172