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Theorem kur14lem6 34271
Description: Lemma for kur14 34276. If π‘˜ is the complementation operator and π‘˜ is the closure operator, this expresses the identity π‘˜π‘π‘˜π΄ = π‘˜π‘π‘˜π‘π‘˜π‘π‘˜π΄ for any subset 𝐴 of the topological space. This is the key result that lets us cut down long enough sequences of π‘π‘˜π‘π‘˜... 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 = βˆͺ 𝐽
kur14lem.k 𝐾 = (clsβ€˜π½)
kur14lem.i 𝐼 = (intβ€˜π½)
kur14lem.a 𝐴 βŠ† 𝑋
kur14lem.b 𝐡 = (𝑋 βˆ– (πΎβ€˜π΄))
Assertion
Ref Expression
kur14lem6 (πΎβ€˜(πΌβ€˜(πΎβ€˜π΅))) = (πΎβ€˜π΅)

Proof of Theorem kur14lem6
StepHypRef Expression
1 kur14lem.j . . . . 5 𝐽 ∈ Top
2 kur14lem.x . . . . . 6 𝑋 = βˆͺ 𝐽
3 kur14lem.k . . . . . 6 𝐾 = (clsβ€˜π½)
4 kur14lem.i . . . . . 6 𝐼 = (intβ€˜π½)
5 kur14lem.b . . . . . . 7 𝐡 = (𝑋 βˆ– (πΎβ€˜π΄))
6 difss 4131 . . . . . . 7 (𝑋 βˆ– (πΎβ€˜π΄)) βŠ† 𝑋
75, 6eqsstri 4016 . . . . . 6 𝐡 βŠ† 𝑋
81, 2, 3, 4, 7kur14lem3 34268 . . . . 5 (πΎβ€˜π΅) βŠ† 𝑋
94fveq1i 6892 . . . . . 6 (πΌβ€˜(πΎβ€˜π΅)) = ((intβ€˜π½)β€˜(πΎβ€˜π΅))
102ntrss2 22568 . . . . . . 7 ((𝐽 ∈ Top ∧ (πΎβ€˜π΅) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(πΎβ€˜π΅)) βŠ† (πΎβ€˜π΅))
111, 8, 10mp2an 690 . . . . . 6 ((intβ€˜π½)β€˜(πΎβ€˜π΅)) βŠ† (πΎβ€˜π΅)
129, 11eqsstri 4016 . . . . 5 (πΌβ€˜(πΎβ€˜π΅)) βŠ† (πΎβ€˜π΅)
132clsss 22565 . . . . 5 ((𝐽 ∈ Top ∧ (πΎβ€˜π΅) βŠ† 𝑋 ∧ (πΌβ€˜(πΎβ€˜π΅)) βŠ† (πΎβ€˜π΅)) β†’ ((clsβ€˜π½)β€˜(πΌβ€˜(πΎβ€˜π΅))) βŠ† ((clsβ€˜π½)β€˜(πΎβ€˜π΅)))
141, 8, 12, 13mp3an 1461 . . . 4 ((clsβ€˜π½)β€˜(πΌβ€˜(πΎβ€˜π΅))) βŠ† ((clsβ€˜π½)β€˜(πΎβ€˜π΅))
153fveq1i 6892 . . . 4 (πΎβ€˜(πΌβ€˜(πΎβ€˜π΅))) = ((clsβ€˜π½)β€˜(πΌβ€˜(πΎβ€˜π΅)))
163fveq1i 6892 . . . 4 (πΎβ€˜(πΎβ€˜π΅)) = ((clsβ€˜π½)β€˜(πΎβ€˜π΅))
1714, 15, 163sstr4i 4025 . . 3 (πΎβ€˜(πΌβ€˜(πΎβ€˜π΅))) βŠ† (πΎβ€˜(πΎβ€˜π΅))
181, 2, 3, 4, 7kur14lem5 34270 . . 3 (πΎβ€˜(πΎβ€˜π΅)) = (πΎβ€˜π΅)
1917, 18sseqtri 4018 . 2 (πΎβ€˜(πΌβ€˜(πΎβ€˜π΅))) βŠ† (πΎβ€˜π΅)
201, 2, 3, 4, 8kur14lem2 34267 . . . . 5 (πΌβ€˜(πΎβ€˜π΅)) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅))))
21 difss 4131 . . . . 5 (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅)))) βŠ† 𝑋
2220, 21eqsstri 4016 . . . 4 (πΌβ€˜(πΎβ€˜π΅)) βŠ† 𝑋
23 kur14lem.a . . . . . . . . 9 𝐴 βŠ† 𝑋
241, 2, 3, 4, 23kur14lem3 34268 . . . . . . . 8 (πΎβ€˜π΄) βŠ† 𝑋
255fveq2i 6894 . . . . . . . . . . 11 (πΎβ€˜π΅) = (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΄)))
2625difeq2i 4119 . . . . . . . . . 10 (𝑋 βˆ– (πΎβ€˜π΅)) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΄))))
271, 2, 3, 4, 24kur14lem2 34267 . . . . . . . . . 10 (πΌβ€˜(πΎβ€˜π΄)) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΄))))
284fveq1i 6892 . . . . . . . . . 10 (πΌβ€˜(πΎβ€˜π΄)) = ((intβ€˜π½)β€˜(πΎβ€˜π΄))
2926, 27, 283eqtr2i 2766 . . . . . . . . 9 (𝑋 βˆ– (πΎβ€˜π΅)) = ((intβ€˜π½)β€˜(πΎβ€˜π΄))
302ntrss2 22568 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (πΎβ€˜π΄) βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(πΎβ€˜π΄)) βŠ† (πΎβ€˜π΄))
311, 24, 30mp2an 690 . . . . . . . . 9 ((intβ€˜π½)β€˜(πΎβ€˜π΄)) βŠ† (πΎβ€˜π΄)
3229, 31eqsstri 4016 . . . . . . . 8 (𝑋 βˆ– (πΎβ€˜π΅)) βŠ† (πΎβ€˜π΄)
332clsss 22565 . . . . . . . 8 ((𝐽 ∈ Top ∧ (πΎβ€˜π΄) βŠ† 𝑋 ∧ (𝑋 βˆ– (πΎβ€˜π΅)) βŠ† (πΎβ€˜π΄)) β†’ ((clsβ€˜π½)β€˜(𝑋 βˆ– (πΎβ€˜π΅))) βŠ† ((clsβ€˜π½)β€˜(πΎβ€˜π΄)))
341, 24, 32, 33mp3an 1461 . . . . . . 7 ((clsβ€˜π½)β€˜(𝑋 βˆ– (πΎβ€˜π΅))) βŠ† ((clsβ€˜π½)β€˜(πΎβ€˜π΄))
353fveq1i 6892 . . . . . . 7 (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅))) = ((clsβ€˜π½)β€˜(𝑋 βˆ– (πΎβ€˜π΅)))
361, 2, 3, 4, 23kur14lem5 34270 . . . . . . . 8 (πΎβ€˜(πΎβ€˜π΄)) = (πΎβ€˜π΄)
373fveq1i 6892 . . . . . . . 8 (πΎβ€˜(πΎβ€˜π΄)) = ((clsβ€˜π½)β€˜(πΎβ€˜π΄))
3836, 37eqtr3i 2762 . . . . . . 7 (πΎβ€˜π΄) = ((clsβ€˜π½)β€˜(πΎβ€˜π΄))
3934, 35, 383sstr4i 4025 . . . . . 6 (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅))) βŠ† (πΎβ€˜π΄)
40 sscon 4138 . . . . . 6 ((πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅))) βŠ† (πΎβ€˜π΄) β†’ (𝑋 βˆ– (πΎβ€˜π΄)) βŠ† (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅)))))
4139, 40ax-mp 5 . . . . 5 (𝑋 βˆ– (πΎβ€˜π΄)) βŠ† (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– (πΎβ€˜π΅))))
4241, 5, 203sstr4i 4025 . . . 4 𝐡 βŠ† (πΌβ€˜(πΎβ€˜π΅))
432clsss 22565 . . . 4 ((𝐽 ∈ Top ∧ (πΌβ€˜(πΎβ€˜π΅)) βŠ† 𝑋 ∧ 𝐡 βŠ† (πΌβ€˜(πΎβ€˜π΅))) β†’ ((clsβ€˜π½)β€˜π΅) βŠ† ((clsβ€˜π½)β€˜(πΌβ€˜(πΎβ€˜π΅))))
441, 22, 42, 43mp3an 1461 . . 3 ((clsβ€˜π½)β€˜π΅) βŠ† ((clsβ€˜π½)β€˜(πΌβ€˜(πΎβ€˜π΅)))
453fveq1i 6892 . . 3 (πΎβ€˜π΅) = ((clsβ€˜π½)β€˜π΅)
4644, 45, 153sstr4i 4025 . 2 (πΎβ€˜π΅) βŠ† (πΎβ€˜(πΌβ€˜(πΎβ€˜π΅)))
4719, 46eqssi 3998 1 (πΎβ€˜(πΌβ€˜(πΎβ€˜π΅))) = (πΎβ€˜π΅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   ∈ wcel 2106   βˆ– cdif 3945   βŠ† wss 3948  βˆͺ cuni 4908  β€˜cfv 6543  Topctop 22402  intcnt 22528  clsccl 22529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22403  df-cld 22530  df-ntr 22531  df-cls 22532
This theorem is referenced by:  kur14lem7  34272
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