Theorem kur14lem6 35205

Description: Lemma for kur14 35210. 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 4102	. . . . . . 7 ⊢ (𝑋 ∖ (𝐾‘𝐴)) ⊆ 𝑋
7	5, 6	eqsstri 3996	. . . . . 6 ⊢ 𝐵 ⊆ 𝑋
8	1, 2, 3, 4, 7	kur14lem3 35202	. . . . 5 ⊢ (𝐾‘𝐵) ⊆ 𝑋
9	4	fveq1i 6862	. . . . . 6 ⊢ (𝐼‘(𝐾‘𝐵)) = ((int‘𝐽)‘(𝐾‘𝐵))
10	2	ntrss2 22951	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵))
11	1, 8, 10	mp2an 692	. . . . . 6 ⊢ ((int‘𝐽)‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
12	9, 11	eqsstri 3996	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ (𝐾‘𝐵)
13	2	clsss 22948	. . . . 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 1463	. . . 4 ⊢ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐵))
15	3	fveq1i 6862	. . . 4 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
16	3	fveq1i 6862	. . . 4 ⊢ (𝐾‘(𝐾‘𝐵)) = ((cls‘𝐽)‘(𝐾‘𝐵))
17	14, 15, 16	3sstr4i 4001	. . 3 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘(𝐾‘𝐵))
18	1, 2, 3, 4, 7	kur14lem5 35204	. . 3 ⊢ (𝐾‘(𝐾‘𝐵)) = (𝐾‘𝐵)
19	17, 18	sseqtri 3998	. 2 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) ⊆ (𝐾‘𝐵)
20	1, 2, 3, 4, 8	kur14lem2 35201	. . . . 5 ⊢ (𝐼‘(𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))))
21		difss 4102	. . . . 5 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐵)))) ⊆ 𝑋
22	20, 21	eqsstri 3996	. . . 4 ⊢ (𝐼‘(𝐾‘𝐵)) ⊆ 𝑋
23		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
24	1, 2, 3, 4, 23	kur14lem3 35202	. . . . . . . 8 ⊢ (𝐾‘𝐴) ⊆ 𝑋
25	5	fveq2i 6864	. . . . . . . . . . 11 ⊢ (𝐾‘𝐵) = (𝐾‘(𝑋 ∖ (𝐾‘𝐴)))
26	25	difeq2i 4089	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝐾‘𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
27	1, 2, 3, 4, 24	kur14lem2 35201	. . . . . . . . . 10 ⊢ (𝐼‘(𝐾‘𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾‘𝐴))))
28	4	fveq1i 6862	. . . . . . . . . 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 22951	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐾‘𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴))
31	1, 24, 30	mp2an 692	. . . . . . . . 9 ⊢ ((int‘𝐽)‘(𝐾‘𝐴)) ⊆ (𝐾‘𝐴)
32	29, 31	eqsstri 3996	. . . . . . . 8 ⊢ (𝑋 ∖ (𝐾‘𝐵)) ⊆ (𝐾‘𝐴)
33	2	clsss 22948	. . . . . . . 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 1463	. . . . . . 7 ⊢ ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ ((cls‘𝐽)‘(𝐾‘𝐴))
35	3	fveq1i 6862	. . . . . . 7 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾‘𝐵)))
36	1, 2, 3, 4, 23	kur14lem5 35204	. . . . . . . 8 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)
37	3	fveq1i 6862	. . . . . . . 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 4001	. . . . . 6 ⊢ (𝐾‘(𝑋 ∖ (𝐾‘𝐵))) ⊆ (𝐾‘𝐴)
40		sscon 4109	. . . . . 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 4001	. . . 4 ⊢ 𝐵 ⊆ (𝐼‘(𝐾‘𝐵))
43	2	clsss 22948	. . . 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 1463	. . 3 ⊢ ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾‘𝐵)))
45	3	fveq1i 6862	. . 3 ⊢ (𝐾‘𝐵) = ((cls‘𝐽)‘𝐵)
46	44, 45, 15	3sstr4i 4001	. 2 ⊢ (𝐾‘𝐵) ⊆ (𝐾‘(𝐼‘(𝐾‘𝐵)))
47	19, 46	eqssi 3966	1 ⊢ (𝐾‘(𝐼‘(𝐾‘𝐵))) = (𝐾‘𝐵)

Syntax hints:   = wceq 1540   ∈ wcel 2109   ∖ cdif 3914   ⊆ wss 3917  ∪ cuni 4874  ‘cfv 6514  Topctop 22787  intcnt 22911  clsccl 22912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-top 22788  df-cld 22913  df-ntr 22914  df-cls 22915

This theorem is referenced by:  kur14lem7  35206