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Theorem ntrin 20913
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 3866 . . . . 5 (𝐴𝐵) ⊆ 𝐴
2 clscld.1 . . . . . 6 𝑋 = 𝐽
32ntrss 20907 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ (𝐴𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
41, 3mp3an3 1453 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
543adant3 1101 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6 inss2 3867 . . . . 5 (𝐴𝐵) ⊆ 𝐵
72ntrss 20907 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋 ∧ (𝐴𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
86, 7mp3an3 1453 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
983adant2 1100 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
105, 9ssind 3870 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11 simp1 1081 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
12 ssinss1 3874 . . . 4 (𝐴𝑋 → (𝐴𝐵) ⊆ 𝑋)
13123ad2ant2 1103 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
142ntropn 20901 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15143adant3 1101 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
162ntropn 20901 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17163adant2 1100 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18 inopn 20752 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
1911, 15, 17, 18syl3anc 1366 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20 inss1 3866 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
212ntrss2 20909 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22213adant3 1101 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
2320, 22syl5ss 3647 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24 inss2 3867 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
252ntrss2 20909 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26253adant2 1100 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
2724, 26syl5ss 3647 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
2823, 27ssind 3870 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴𝐵))
292ssntr 20910 . . 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 1367 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴𝐵)))
3110, 30eqssd 3653 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  cin 3606  wss 3607   cuni 4468  cfv 5926  Topctop 20746  intcnt 20869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by:  dvreslem  23718  dvaddbr  23746  dvmulbr  23747  clsun  32448  neiin  32452
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