Theorem neitx 21410

Description: The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)

Hypotheses

Ref	Expression
neitx.x	⊢ 𝑋 = ∪ 𝐽
neitx.y	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
neitx	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))

Proof of Theorem neitx

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neitx.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	neii1 20910	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴 ⊆ 𝑋)
3	2	ad2ant2r 783	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴 ⊆ 𝑋)
4		neitx.y	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	4	neii1 20910	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵 ⊆ 𝑌)
6	5	ad2ant2l 782	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵 ⊆ 𝑌)
7		xpss12 5225	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
8	3, 6, 7	syl2anc 693	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
9	1, 4	txuni 21395	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
10	9	adantr 481	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
11	8, 10	sseqtrd 3641	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ ∪ (𝐽 ×t 𝐾))
12		simp-5l 808	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
13		simp-4r 807	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → 𝑎 ∈ 𝐽)
14		simplr 792	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → 𝑏 ∈ 𝐾)
15		txopn 21405	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
16	12, 13, 14, 15	syl12anc 1324	. . . . 5 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
17		simpr1l 1118	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ ((𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴) ∧ 𝑏 ∈ 𝐾 ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))) → 𝐶 ⊆ 𝑎)
18	17	3anassrs 1290	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → 𝐶 ⊆ 𝑎)
19		simprl 794	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → 𝐷 ⊆ 𝑏)
20		xpss12 5225	. . . . . 6 ⊢ ((𝐶 ⊆ 𝑎 ∧ 𝐷 ⊆ 𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
21	18, 19, 20	syl2anc 693	. . . . 5 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
22		simpr1r 1119	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ ((𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴) ∧ 𝑏 ∈ 𝐾 ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))) → 𝑎 ⊆ 𝐴)
23	22	3anassrs 1290	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → 𝑎 ⊆ 𝐴)
24		simprr 796	. . . . . 6 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → 𝑏 ⊆ 𝐵)
25		xpss12 5225	. . . . . 6 ⊢ ((𝑎 ⊆ 𝐴 ∧ 𝑏 ⊆ 𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
26	23, 24, 25	syl2anc 693	. . . . 5 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
27		sseq2 3627	. . . . . . 7 ⊢ (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28		sseq1 3626	. . . . . . 7 ⊢ (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
29	27, 28	anbi12d 747	. . . . . 6 ⊢ (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
30	29	rspcev 3309	. . . . 5 ⊢ (((𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾) ∧ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)))
31	16, 21, 26, 30	syl12anc 1324	. . . 4 ⊢ (((((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) ∧ 𝑏 ∈ 𝐾) ∧ (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵)) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)))
32		neii2 20912	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
33	32	ad2ant2l 782	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
34	33	ad2antrr 762	. . . 4 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
35	31, 34	r19.29a 3078	. . 3 ⊢ (((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) ∧ 𝑎 ∈ 𝐽) ∧ (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴)) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)))
36		neii2 20912	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎 ∈ 𝐽 (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴))
37	36	ad2ant2r 783	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎 ∈ 𝐽 (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴))
38	35, 37	r19.29a 3078	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)))
39		txtop 21372	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
40	39	adantr 481	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
41	1	neiss2 20905	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶 ⊆ 𝑋)
42	41	ad2ant2r 783	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶 ⊆ 𝑋)
43	4	neiss2 20905	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷 ⊆ 𝑌)
44	43	ad2ant2l 782	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷 ⊆ 𝑌)
45		xpss12 5225	. . . . 5 ⊢ ((𝐶 ⊆ 𝑋 ∧ 𝐷 ⊆ 𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
46	42, 44, 45	syl2anc 693	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
47	46, 10	sseqtrd 3641	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ ∪ (𝐽 ×t 𝐾))
48		eqid 2622	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
49	48	isnei 20907	. . 3 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ (𝐶 × 𝐷) ⊆ ∪ (𝐽 ×t 𝐾)) → ((𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)) ↔ ((𝐴 × 𝐵) ⊆ ∪ (𝐽 ×t 𝐾) ∧ ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)))))
50	40, 47, 49	syl2anc 693	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ((𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)) ↔ ((𝐴 × 𝐵) ⊆ ∪ (𝐽 ×t 𝐾) ∧ ∃𝑐 ∈ (𝐽 ×t 𝐾)((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)))))
51	11, 38, 50	mpbir2and 957	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ⊆ wss 3574  ∪ cuni 4436   × cxp 5112  ‘cfv 5888  (class class class)co 6650  Topctop 20698  neicnei 20901   ×_t ctx 21363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-nei 20902  df-tx 21365

This theorem is referenced by:  utop2nei  22054  utop3cls  22055