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Theorem ntrin 12330
Description: A pairwise intersection of interiors is the interior of the intersection. This does not always hold for arbitrary intersections. (Contributed by Jeff Hankins, 31-Aug-2009.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
ntrin ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))

Proof of Theorem ntrin
StepHypRef Expression
1 inss1 3300 . . . . 5 (𝐴𝐵) ⊆ 𝐴
2 clscld.1 . . . . . 6 𝑋 = 𝐽
32ntrss 12325 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ (𝐴𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
41, 3mp3an3 1305 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
543adant3 1002 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6 inss2 3301 . . . . 5 (𝐴𝐵) ⊆ 𝐵
72ntrss 12325 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋 ∧ (𝐴𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
86, 7mp3an3 1305 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
983adant2 1001 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
105, 9ssind 3304 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11 simp1 982 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
12 ssinss1 3309 . . . 4 (𝐴𝑋 → (𝐴𝐵) ⊆ 𝑋)
13123ad2ant2 1004 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
142ntropn 12323 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15143adant3 1002 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
162ntropn 12323 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17163adant2 1001 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18 inopn 12207 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
1911, 15, 17, 18syl3anc 1217 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20 inss1 3300 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
212ntrss2 12327 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22213adant3 1002 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
2320, 22sstrid 3112 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24 inss2 3301 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
252ntrss2 12327 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26253adant2 1001 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
2724, 26sstrid 3112 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
2823, 27ssind 3304 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴𝐵))
292ssntr 12328 . . 3 (((𝐽 ∈ Top ∧ (𝐴𝐵) ⊆ 𝑋) ∧ ((((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽 ∧ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴𝐵))) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴𝐵)))
3011, 13, 19, 28, 29syl22anc 1218 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴𝐵)))
3110, 30eqssd 3118 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 963   = wceq 1332  wcel 1481  cin 3074  wss 3075   cuni 3743  cfv 5130  Topctop 12201  intcnt 12299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-top 12202  df-ntr 12302
This theorem is referenced by: (None)
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