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Theorem ntrin 21086
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 3981 . . . . 5 (𝐴𝐵) ⊆ 𝐴
2 clscld.1 . . . . . 6 𝑋 = 𝐽
32ntrss 21080 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋 ∧ (𝐴𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
41, 3mp3an3 1561 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
543adant3 1126 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6 inss2 3982 . . . . 5 (𝐴𝐵) ⊆ 𝐵
72ntrss 21080 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋 ∧ (𝐴𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
86, 7mp3an3 1561 . . . 4 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
983adant2 1125 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ ((int‘𝐽)‘𝐵))
105, 9ssind 3985 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11 simp1 1130 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → 𝐽 ∈ Top)
12 ssinss1 3990 . . . 4 (𝐴𝑋 → (𝐴𝐵) ⊆ 𝑋)
13123ad2ant2 1128 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐵) ⊆ 𝑋)
142ntropn 21074 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15143adant3 1126 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
162ntropn 21074 . . . . 5 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17163adant2 1125 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18 inopn 20924 . . . 4 ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
1911, 15, 17, 18syl3anc 1476 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20 inss1 3981 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
212ntrss2 21082 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22213adant3 1126 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
2320, 22syl5ss 3763 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24 inss2 3982 . . . . 5 (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
252ntrss2 21082 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26253adant2 1125 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
2724, 26syl5ss 3763 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
2823, 27ssind 3985 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴𝐵))
292ssntr 21083 . . 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 1477 . 2 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴𝐵)))
3110, 30eqssd 3769 1 ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((int‘𝐽)‘(𝐴𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  cin 3722  wss 3723   cuni 4575  cfv 6030  Topctop 20918  intcnt 21042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-top 20919  df-cld 21044  df-ntr 21045  df-cls 21046
This theorem is referenced by:  dvreslem  23893  dvaddbr  23921  dvmulbr  23922  clsun  32660  neiin  32664
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