Theorem ntrin 14851

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 3427	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2		clscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	ntrss 14846	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐴))
4	1, 3	mp3an3 1362	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐴))
5	4	3adant3 1043	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐴))
6		inss2 3428	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
7	2	ntrss 14846	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐵))
8	6, 7	mp3an3 1362	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐵))
9	8	3adant2 1042	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ ((int‘𝐽)‘𝐵))
10	5, 9	ssind 3431	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) ⊆ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))
11		simp1 1023	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → 𝐽 ∈ Top)
12		ssinss1 3436	. . . 4 ⊢ (𝐴 ⊆ 𝑋 → (𝐴 ∩ 𝐵) ⊆ 𝑋)
13	12	3ad2ant2 1045	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
14	2	ntropn 14844	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
15	14	3adant3 1043	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ∈ 𝐽)
16	2	ntropn 14844	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
17	16	3adant2 1042	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ∈ 𝐽)
18		inopn 14730	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝐴) ∈ 𝐽 ∧ ((int‘𝐽)‘𝐵) ∈ 𝐽) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
19	11, 15, 17, 18	syl3anc 1273	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ∈ 𝐽)
20		inss1 3427	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐴)
21	2	ntrss2 14848	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
22	21	3adant3 1043	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) ⊆ 𝐴)
23	20, 22	sstrid 3238	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐴)
24		inss2 3428	. . . . 5 ⊢ (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘𝐵)
25	2	ntrss2 14848	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
26	25	3adant2 1042	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘𝐵) ⊆ 𝐵)
27	24, 26	sstrid 3238	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ 𝐵)
28	23, 27	ssind 3431	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ (𝐴 ∩ 𝐵))
29	2	ssntr 14849	. . 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 1274	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)) ⊆ ((int‘𝐽)‘(𝐴 ∩ 𝐵)))
31	10, 30	eqssd 3244	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((int‘𝐽)‘(𝐴 ∩ 𝐵)) = (((int‘𝐽)‘𝐴) ∩ ((int‘𝐽)‘𝐵)))

Syntax hints:   → wi 4   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202   ∩ cin 3199   ⊆ wss 3200  ∪ cuni 3893  ‘cfv 5326  Topctop 14724  intcnt 14820

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-top 14725  df-ntr 14823

This theorem is referenced by: (None)