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Theorem kur14lem6 35181
Description: Lemma for kur14 35186. 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 4159 . . . . . . 7 (𝑋 ∖ (𝐾𝐴)) ⊆ 𝑋
75, 6eqsstri 4043 . . . . . 6 𝐵𝑋
81, 2, 3, 4, 7kur14lem3 35178 . . . . 5 (𝐾𝐵) ⊆ 𝑋
94fveq1i 6923 . . . . . 6 (𝐼‘(𝐾𝐵)) = ((int‘𝐽)‘(𝐾𝐵))
102ntrss2 23088 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝐾𝐵) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾𝐵)) ⊆ (𝐾𝐵))
111, 8, 10mp2an 691 . . . . . 6 ((int‘𝐽)‘(𝐾𝐵)) ⊆ (𝐾𝐵)
129, 11eqsstri 4043 . . . . 5 (𝐼‘(𝐾𝐵)) ⊆ (𝐾𝐵)
132clsss 23085 . . . . 5 ((𝐽 ∈ Top ∧ (𝐾𝐵) ⊆ 𝑋 ∧ (𝐼‘(𝐾𝐵)) ⊆ (𝐾𝐵)) → ((cls‘𝐽)‘(𝐼‘(𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐵)))
141, 8, 12, 13mp3an 1461 . . . 4 ((cls‘𝐽)‘(𝐼‘(𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐵))
153fveq1i 6923 . . . 4 (𝐾‘(𝐼‘(𝐾𝐵))) = ((cls‘𝐽)‘(𝐼‘(𝐾𝐵)))
163fveq1i 6923 . . . 4 (𝐾‘(𝐾𝐵)) = ((cls‘𝐽)‘(𝐾𝐵))
1714, 15, 163sstr4i 4052 . . 3 (𝐾‘(𝐼‘(𝐾𝐵))) ⊆ (𝐾‘(𝐾𝐵))
181, 2, 3, 4, 7kur14lem5 35180 . . 3 (𝐾‘(𝐾𝐵)) = (𝐾𝐵)
1917, 18sseqtri 4045 . 2 (𝐾‘(𝐼‘(𝐾𝐵))) ⊆ (𝐾𝐵)
201, 2, 3, 4, 8kur14lem2 35177 . . . . 5 (𝐼‘(𝐾𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐵))))
21 difss 4159 . . . . 5 (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐵)))) ⊆ 𝑋
2220, 21eqsstri 4043 . . . 4 (𝐼‘(𝐾𝐵)) ⊆ 𝑋
23 kur14lem.a . . . . . . . . 9 𝐴𝑋
241, 2, 3, 4, 23kur14lem3 35178 . . . . . . . 8 (𝐾𝐴) ⊆ 𝑋
255fveq2i 6925 . . . . . . . . . . 11 (𝐾𝐵) = (𝐾‘(𝑋 ∖ (𝐾𝐴)))
2625difeq2i 4146 . . . . . . . . . 10 (𝑋 ∖ (𝐾𝐵)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐴))))
271, 2, 3, 4, 24kur14lem2 35177 . . . . . . . . . 10 (𝐼‘(𝐾𝐴)) = (𝑋 ∖ (𝐾‘(𝑋 ∖ (𝐾𝐴))))
284fveq1i 6923 . . . . . . . . . 10 (𝐼‘(𝐾𝐴)) = ((int‘𝐽)‘(𝐾𝐴))
2926, 27, 283eqtr2i 2774 . . . . . . . . 9 (𝑋 ∖ (𝐾𝐵)) = ((int‘𝐽)‘(𝐾𝐴))
302ntrss2 23088 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐾𝐴) ⊆ 𝑋) → ((int‘𝐽)‘(𝐾𝐴)) ⊆ (𝐾𝐴))
311, 24, 30mp2an 691 . . . . . . . . 9 ((int‘𝐽)‘(𝐾𝐴)) ⊆ (𝐾𝐴)
3229, 31eqsstri 4043 . . . . . . . 8 (𝑋 ∖ (𝐾𝐵)) ⊆ (𝐾𝐴)
332clsss 23085 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐾𝐴) ⊆ 𝑋 ∧ (𝑋 ∖ (𝐾𝐵)) ⊆ (𝐾𝐴)) → ((cls‘𝐽)‘(𝑋 ∖ (𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐴)))
341, 24, 32, 33mp3an 1461 . . . . . . 7 ((cls‘𝐽)‘(𝑋 ∖ (𝐾𝐵))) ⊆ ((cls‘𝐽)‘(𝐾𝐴))
353fveq1i 6923 . . . . . . 7 (𝐾‘(𝑋 ∖ (𝐾𝐵))) = ((cls‘𝐽)‘(𝑋 ∖ (𝐾𝐵)))
361, 2, 3, 4, 23kur14lem5 35180 . . . . . . . 8 (𝐾‘(𝐾𝐴)) = (𝐾𝐴)
373fveq1i 6923 . . . . . . . 8 (𝐾‘(𝐾𝐴)) = ((cls‘𝐽)‘(𝐾𝐴))
3836, 37eqtr3i 2770 . . . . . . 7 (𝐾𝐴) = ((cls‘𝐽)‘(𝐾𝐴))
3934, 35, 383sstr4i 4052 . . . . . 6 (𝐾‘(𝑋 ∖ (𝐾𝐵))) ⊆ (𝐾𝐴)
40 sscon 4166 . . . . . 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 4052 . . . 4 𝐵 ⊆ (𝐼‘(𝐾𝐵))
432clsss 23085 . . . 4 ((𝐽 ∈ Top ∧ (𝐼‘(𝐾𝐵)) ⊆ 𝑋𝐵 ⊆ (𝐼‘(𝐾𝐵))) → ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾𝐵))))
441, 22, 42, 43mp3an 1461 . . 3 ((cls‘𝐽)‘𝐵) ⊆ ((cls‘𝐽)‘(𝐼‘(𝐾𝐵)))
453fveq1i 6923 . . 3 (𝐾𝐵) = ((cls‘𝐽)‘𝐵)
4644, 45, 153sstr4i 4052 . 2 (𝐾𝐵) ⊆ (𝐾‘(𝐼‘(𝐾𝐵)))
4719, 46eqssi 4025 1 (𝐾‘(𝐼‘(𝐾𝐵))) = (𝐾𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  cdif 3973  wss 3976   cuni 4931  cfv 6575  Topctop 22922  intcnt 23048  clsccl 23049
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-top 22923  df-cld 23050  df-ntr 23051  df-cls 23052
This theorem is referenced by:  kur14lem7  35182
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