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Theorem ntrin 14109
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 3370 . . . . 5 (𝐴𝐵) ⊆ 𝐴
2 clscld.1 . . . . . 6 𝑋 = 𝐽
32ntrss 14104 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ (𝐴𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
41, 3mp3an3 1337 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
543adant3 1019 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6 inss2 3371 . . . . 5 (𝐴𝐵) ⊆ 𝐵
72ntrss 14104 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋 ∧ (𝐴𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
86, 7mp3an3 1337 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
983adant2 1018 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
105, 9ssind 3374 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11 simp1 999 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
12 ssinss1 3379 . . . 4 (𝐴𝑋 → (𝐴𝐵) ⊆ 𝑋)
13123ad2ant2 1021 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
142ntropn 14102 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15143adant3 1019 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
162ntropn 14102 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17163adant2 1018 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18 inopn 13988 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
1911, 15, 17, 18syl3anc 1249 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20 inss1 3370 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
212ntrss2 14106 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22213adant3 1019 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
2320, 22sstrid 3181 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24 inss2 3371 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
252ntrss2 14106 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26253adant2 1018 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
2724, 26sstrid 3181 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
2823, 27ssind 3374 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴𝐵))
292ssntr 14107 . . 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 1250 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴𝐵)))
3110, 30eqssd 3187 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 980   = wceq 1364  wcel 2160  cin 3143  wss 3144   cuni 3827  cfv 5238  Topctop 13982  intcnt 14078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-top 13983  df-ntr 14081
This theorem is referenced by: (None)
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