Theorem neitx 15133

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 15012	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐴 ⊆ 𝑋)
3	2	ad2ant2r 509	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐴 ⊆ 𝑋)
4		neitx.y	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	4	neii1 15012	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐵 ⊆ 𝑌)
6	5	ad2ant2l 508	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐵 ⊆ 𝑌)
7		xpss12 4857	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
8	3, 6, 7	syl2anc 411	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ (𝑋 × 𝑌))
9	1, 4	txuni 15128	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
10	9	adantr 276	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
11	8, 10	sseqtrd 3276	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ⊆ ∪ (𝐽 ×t 𝐾))
12		simp-5l 545	. . . . . 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 544	. . . . . 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 529	. . . . . 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 15130	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑎 ∈ 𝐽 ∧ 𝑏 ∈ 𝐾)) → (𝑎 × 𝑏) ∈ (𝐽 ×t 𝐾))
16	12, 13, 14, 15	syl12anc 1272	. . . . 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 1081	. . . . . . 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 1256	. . . . . 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 531	. . . . . 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 4857	. . . . . 6 ⊢ ((𝐶 ⊆ 𝑎 ∧ 𝐷 ⊆ 𝑏) → (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏))
21	18, 19, 20	syl2anc 411	. . . . 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 1082	. . . . . . 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 1256	. . . . . 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 533	. . . . . 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 4857	. . . . . 6 ⊢ ((𝑎 ⊆ 𝐴 ∧ 𝑏 ⊆ 𝐵) → (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))
26	23, 24, 25	syl2anc 411	. . . . 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 3262	. . . . . . 7 ⊢ (𝑐 = (𝑎 × 𝑏) → ((𝐶 × 𝐷) ⊆ 𝑐 ↔ (𝐶 × 𝐷) ⊆ (𝑎 × 𝑏)))
28		sseq1 3261	. . . . . . 7 ⊢ (𝑐 = (𝑎 × 𝑏) → (𝑐 ⊆ (𝐴 × 𝐵) ↔ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵)))
29	27, 28	anbi12d 473	. . . . . 6 ⊢ (𝑐 = (𝑎 × 𝑏) → (((𝐶 × 𝐷) ⊆ 𝑐 ∧ 𝑐 ⊆ (𝐴 × 𝐵)) ↔ ((𝐶 × 𝐷) ⊆ (𝑎 × 𝑏) ∧ (𝑎 × 𝑏) ⊆ (𝐴 × 𝐵))))
30	29	rspcev 2921	. . . . 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 1272	. . . 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 15014	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
33	32	ad2ant2l 508	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑏 ∈ 𝐾 (𝐷 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝐵))
34	33	ad2antrr 488	. . . 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 2686	. . 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 15014	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → ∃𝑎 ∈ 𝐽 (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴))
37	36	ad2ant2r 509	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → ∃𝑎 ∈ 𝐽 (𝐶 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝐴))
38	35, 37	r19.29a 2686	. 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 15125	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
40	39	adantr 276	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐽 ×t 𝐾) ∈ Top)
41	1	neiss2 15007	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ ((nei‘𝐽)‘𝐶)) → 𝐶 ⊆ 𝑋)
42	41	ad2ant2r 509	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐶 ⊆ 𝑋)
43	4	neiss2 15007	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷)) → 𝐷 ⊆ 𝑌)
44	43	ad2ant2l 508	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → 𝐷 ⊆ 𝑌)
45		xpss12 4857	. . . . 5 ⊢ ((𝐶 ⊆ 𝑋 ∧ 𝐷 ⊆ 𝑌) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
46	42, 44, 45	syl2anc 411	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ (𝑋 × 𝑌))
47	46, 10	sseqtrd 3276	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐶 × 𝐷) ⊆ ∪ (𝐽 ×t 𝐾))
48		eqid 2232	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
49	48	isnei 15009	. . 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 411	. 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 953	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   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∃wrex 2521   ⊆ wss 3211  ∪ cuni 3914   × cxp 4747  ‘cfv 5352  (class class class)co 6050  Topctop 14862  neicnei 15003   ×_t ctx 15117

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-topgen 13473  df-top 14863  df-topon 14876  df-bases 14908  df-nei 15004  df-tx 15118

This theorem is referenced by: (None)