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Theorem kur14lem6 31735
Description: Lemma for kur14 31740. 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 𝐴𝑋
kur14lem.b 𝐵 = (𝑋 ∖ (𝐾𝐴))
Assertion
Ref Expression
kur14lem6 (𝐾‘(𝐼‘(𝐾𝐵))) = (𝐾𝐵)

Proof of Theorem kur14lem6
StepHypRef 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 ∖ (𝐾𝐴))
6 difss 3966 . . . . . . 7 (𝑋 ∖ (𝐾𝐴)) ⊆ 𝑋
75, 6eqsstri 3860 . . . . . 6 𝐵𝑋
81, 2, 3, 4, 7kur14lem3 31732 . . . . 5 (𝐾𝐵) ⊆ 𝑋
94fveq1i 6438 . . . . . 6 (𝐼‘(𝐾𝐵)) = ((int‘𝐽)‘(𝐾𝐵))
102ntrss2 21239 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐾𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾𝐵)) ⊆ (𝐾𝐵))
111, 8, 10mp2an 683 . . . . . 6 ((int‘𝐽)‘(𝐾𝐵)) ⊆ (𝐾𝐵)
129, 11eqsstri 3860 . . . . 5 (𝐼‘(𝐾𝐵)) ⊆ (𝐾𝐵)
132clsss 21236 . . . . 5 ((𝐽 ∈ Top ∧ (𝐾𝐵) ⊆ 𝑋 ∧ (𝐼‘(𝐾𝐵)) ⊆ (𝐾𝐵)) → ((cls‘𝐽)‘(𝐼‘(𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐵)))
141, 8, 12, 13mp3an 1589 . . . 4 ((cls‘𝐽)‘(𝐼‘(𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐵))
153fveq1i 6438 . . . 4 (𝐾‘(𝐼‘(𝐾𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾𝐵)))
163fveq1i 6438 . . . 4 (𝐾‘(𝐾𝐵)) = ((cls‘𝐽)‘(𝐾𝐵))
1714, 15, 163sstr4i 3869 . . 3 (𝐾‘(𝐼‘(𝐾𝐵))) ⊆ (𝐾‘(𝐾𝐵))
181, 2, 3, 4, 7kur14lem5 31734 . . 3 (𝐾‘(𝐾𝐵)) = (𝐾𝐵)
1917, 18sseqtri 3862 . 2 (𝐾‘(𝐼‘(𝐾𝐵))) ⊆ (𝐾𝐵)
201, 2, 3, 4, 8kur14lem2 31731 . . . . 5 (𝐼‘(𝐾𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐵))))
21 difss 3966 . . . . 5 (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐵)))) ⊆ 𝑋
2220, 21eqsstri 3860 . . . 4 (𝐼‘(𝐾𝐵)) ⊆ 𝑋
23 kur14lem.a . . . . . . . . 9 𝐴𝑋
241, 2, 3, 4, 23kur14lem3 31732 . . . . . . . 8 (𝐾𝐴) ⊆ 𝑋
255fveq2i 6440 . . . . . . . . . . 11 (𝐾𝐵) = (𝐾‘(𝑋 ∖ (𝐾𝐴)))
2625difeq2i 3954 . . . . . . . . . 10 (𝑋 ∖ (𝐾𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐴))))
271, 2, 3, 4, 24kur14lem2 31731 . . . . . . . . . 10 (𝐼‘(𝐾𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐴))))
284fveq1i 6438 . . . . . . . . . 10 (𝐼‘(𝐾𝐴)) = ((int‘𝐽)‘(𝐾𝐴))
2926, 27, 283eqtr2i 2855 . . . . . . . . 9 (𝑋 ∖ (𝐾𝐵)) = ((int‘𝐽)‘(𝐾𝐴))
302ntrss2 21239 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐾𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾𝐴)) ⊆ (𝐾𝐴))
311, 24, 30mp2an 683 . . . . . . . . 9 ((int‘𝐽)‘(𝐾𝐴)) ⊆ (𝐾𝐴)
3229, 31eqsstri 3860 . . . . . . . 8 (𝑋 ∖ (𝐾𝐵)) ⊆ (𝐾𝐴)
332clsss 21236 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐾𝐴) ⊆ 𝑋 ∧ (𝑋 ∖ (𝐾𝐵)) ⊆ (𝐾𝐴)) → ((cls‘𝐽)‘(𝑋 ∖ (𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐴)))
341, 24, 32, 33mp3an 1589 . . . . . . 7 ((cls‘𝐽)‘(𝑋 ∖ (𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐴))
353fveq1i 6438 . . . . . . 7 (𝐾‘(𝑋 ∖ (𝐾𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾𝐵)))
361, 2, 3, 4, 23kur14lem5 31734 . . . . . . . 8 (𝐾‘(𝐾𝐴)) = (𝐾𝐴)
373fveq1i 6438 . . . . . . . 8 (𝐾‘(𝐾𝐴)) = ((cls‘𝐽)‘(𝐾𝐴))
3836, 37eqtr3i 2851 . . . . . . 7 (𝐾𝐴) = ((cls‘𝐽)‘(𝐾𝐴))
3934, 35, 383sstr4i 3869 . . . . . 6 (𝐾‘(𝑋 ∖ (𝐾𝐵))) ⊆ (𝐾𝐴)
40 sscon 3973 . . . . . 6 ((𝐾‘(𝑋 ∖ (𝐾𝐵))) ⊆ (𝐾𝐴) → (𝑋 ∖ (𝐾𝐴)) ⊆ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐵)))))
4139, 40ax-mp 5 . . . . 5 (𝑋 ∖ (𝐾𝐴)) ⊆ (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐵))))
4241, 5, 203sstr4i 3869 . . . 4 𝐵 ⊆ (𝐼‘(𝐾𝐵))
432clsss 21236 . . . 4 ((𝐽 ∈ Top ∧ (𝐼‘(𝐾𝐵)) ⊆ 𝑋𝐵 ⊆ (𝐼‘(𝐾𝐵))) → ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾𝐵))))
441, 22, 42, 43mp3an 1589 . . 3 ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾𝐵)))
453fveq1i 6438 . . 3 (𝐾𝐵) = ((cls‘𝐽)‘𝐵)
4644, 45, 153sstr4i 3869 . 2 (𝐾𝐵) ⊆ (𝐾‘(𝐼‘(𝐾𝐵)))
4719, 46eqssi 3843 1 (𝐾‘(𝐼‘(𝐾𝐵))) = (𝐾𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1656  wcel 2164  cdif 3795  wss 3798   cuni 4660  cfv 6127  Topctop 21075  intcnt 21199  clsccl 21200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-top 21076  df-cld 21201  df-ntr 21202  df-cls 21203
This theorem is referenced by:  kur14lem7  31736
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