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Theorem ntrin 21673
 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 4155 . . . . 5 (𝐴𝐵) ⊆ 𝐴
2 clscld.1 . . . . . 6 𝑋 = 𝐽
32ntrss 21667 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ (𝐴𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
41, 3mp3an3 1447 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
543adant3 1129 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6 inss2 4156 . . . . 5 (𝐴𝐵) ⊆ 𝐵
72ntrss 21667 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋 ∧ (𝐴𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
86, 7mp3an3 1447 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
983adant2 1128 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
105, 9ssind 4159 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11 simp1 1133 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
12 ssinss1 4164 . . . 4 (𝐴𝑋 → (𝐴𝐵) ⊆ 𝑋)
13123ad2ant2 1131 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
142ntropn 21661 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15143adant3 1129 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
162ntropn 21661 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17163adant2 1128 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18 inopn 21511 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
1911, 15, 17, 18syl3anc 1368 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20 inss1 4155 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
212ntrss2 21669 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22213adant3 1129 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
2320, 22sstrid 3926 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24 inss2 4156 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
252ntrss2 21669 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26253adant2 1128 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
2724, 26sstrid 3926 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
2823, 27ssind 4159 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴𝐵))
292ssntr 21670 . . 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 837 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴𝐵)))
3110, 30eqssd 3932 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3880   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324  Topctop 21505  intcnt 21629 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21506  df-cld 21631  df-ntr 21632  df-cls 21633 This theorem is referenced by:  dvreslem  24519  dvaddbr  24548  dvmulbr  24549  clsun  33801  neiin  33805
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