Theorem neitx 21220

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 20720	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴 ⊆ 𝑋)
3	2	ad2ant2r 779	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴 ⊆ 𝑋)
4		neitx.y	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	4	neii1 20720	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵 ⊆ 𝑌)
6	5	ad2ant2l 778	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵 ⊆ 𝑌)
7		xpss12 5148	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
8	3, 6, 7	syl2anc 691	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
9	1, 4	txuni 21205	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
10	9	adantr 480	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
11	8, 10	sseqtrd 3604	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ ∪ (𝐽 ×t 𝐾))
12		simp-5l 804	. . . . . 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 803	. . . . . 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 788	. . . . . 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 21215	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
16	12, 13, 14, 15	syl12anc 1316	. . . . 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 1111	. . . . . . 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 1282	. . . . . 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 790	. . . . . 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 5148	. . . . . 6 ⊢ ((𝐶 ⊆ 𝑎 ∧ 𝐷 ⊆ 𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
21	18, 19, 20	syl2anc 691	. . . . 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 1112	. . . . . . 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 1282	. . . . . 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 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 ⊆ 𝐵)
25		xpss12 5148	. . . . . 6 ⊢ ((𝑎 ⊆ 𝐴 ∧ 𝑏 ⊆ 𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
26	23, 24, 25	syl2anc 691	. . . . 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 3590	. . . . . . 7 ⊢ (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28		sseq1 3589	. . . . . . 7 ⊢ (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
29	27, 28	anbi12d 743	. . . . . 6 ⊢ (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
30	29	rspcev 3282	. . . . 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 1316	. . . 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 20722	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
33	32	ad2ant2l 778	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
34	33	ad2antrr 758	. . . 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 3060	. . 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 20722	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎 ∈ 𝐽 (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴))
37	36	ad2ant2r 779	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎 ∈ 𝐽 (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴))
38	35, 37	r19.29a 3060	. 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 21182	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
40	39	adantr 480	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
41	1	neiss2 20715	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶 ⊆ 𝑋)
42	41	ad2ant2r 779	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶 ⊆ 𝑋)
43	4	neiss2 20715	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷 ⊆ 𝑌)
44	43	ad2ant2l 778	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷 ⊆ 𝑌)
45		xpss12 5148	. . . . 5 ⊢ ((𝐶 ⊆ 𝑋 ∧ 𝐷 ⊆ 𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
46	42, 44, 45	syl2anc 691	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
47	46, 10	sseqtrd 3604	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ ∪ (𝐽 ×t 𝐾))
48		eqid 2610	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
49	48	isnei 20717	. . 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 691	. 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 959	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ∪ cuni 4372   × cxp 5036  ‘cfv 5804  (class class class)co 6549  Topctop 20517  neicnei 20711   ×_t ctx 21173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-nei 20712  df-tx 21175

This theorem is referenced by:  utop2nei  21864  utop3cls  21865