Theorem ntrin 14711

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 ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))

Proof of Theorem ntrin

Step	Hyp	Ref	Expression
1		inss1 3401	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		clscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	ntrss 14706	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐴))
4	1, 3	mp3an3 1339	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐴))
5	4	3adant3 1020	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6		inss2 3402	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	2	ntrss 14706	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐵))
8	6, 7	mp3an3 1339	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐵))
9	8	3adant2 1019	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐵))
10	5, 9	ssind 3405	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11		simp1 1000	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐽 ∈ Top)
12		ssinss1 3410	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ 𝐵) ⊆ 𝑋)
13	12	3ad2ant2 1022	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
14	2	ntropn 14704	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15	14	3adant3 1020	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
16	2	ntropn 14704	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17	16	3adant2 1019	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18		inopn 14590	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
19	11, 15, 17, 18	syl3anc 1250	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20		inss1 3401	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
21	2	ntrss2 14708	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22	21	3adant3 1020	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
23	20, 22	sstrid 3212	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24		inss2 3402	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
25	2	ntrss2 14708	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26	25	3adant2 1019	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
27	24, 26	sstrid 3212	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
28	23, 27	ssind 3405	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴 ∩ 𝐵))
29	2	ssntr 14709	. . 3 ⊢ (((𝐽 ∈ Top ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) ∧ ((((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽 ∧ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴 ∩ 𝐵))) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴 ∩ 𝐵)))
30	11, 13, 19, 28, 29	syl22anc 1251	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴 ∩ 𝐵)))
31	10, 30	eqssd 3218	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))

Syntax hints:   → wi 4   ∧ w3a 981   = wceq 1373   ∈ wcel 2178   ∩ cin 3173   ⊆ wss 3174  ∪ cuni 3864  ‘cfv 5290  Topctop 14584  intcnt 14680

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-top 14585  df-ntr 14683

This theorem is referenced by: (None)