Theorem xmettx 14678

Description: The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)

Hypotheses

Ref	Expression
xmetxp.p	⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))
xmetxp.1	⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
xmetxp.2	⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
xmettx.j	⊢ 𝐽 = (MetOpen‘𝑀)
xmettx.k	⊢ 𝐾 = (MetOpen‘𝑁)
xmettx.l	⊢ 𝐿 = (MetOpen‘𝑃)

Assertion

Ref	Expression
xmettx	⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾))

Distinct variable groups:   𝑢,𝑀,𝑣   𝑢,𝑁,𝑣   𝑢,𝑋,𝑣   𝑢,𝑌,𝑣

Allowed substitution hints:   𝜑(𝑣,𝑢)   𝑃(𝑣,𝑢)   𝐽(𝑣,𝑢)   𝐾(𝑣,𝑢)   𝐿(𝑣,𝑢)

Proof of Theorem xmettx

Dummy variables 𝑗 𝑘 𝑚 𝑛 𝑥 𝑦 𝑟 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.p	. . 3 ⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))
2		xmetxp.1	. . 3 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
3		xmetxp.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
4		xmettx.j	. . 3 ⊢ 𝐽 = (MetOpen‘𝑀)
5		xmettx.k	. . 3 ⊢ 𝐾 = (MetOpen‘𝑁)
6		xmettx.l	. . 3 ⊢ 𝐿 = (MetOpen‘𝑃)
7	1, 2, 3, 4, 5, 6	xmettxlem 14677	. 2 ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))
8		eqid 2193	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
9	8	elrnmpog 6031	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠)))
10	9	elv 2764	. . . . . . . . . 10 ⊢ (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
11	10	biimpi 120	. . . . . . . . 9 ⊢ (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
12	11	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) → ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠))
13		xpeq1 4673	. . . . . . . . . 10 ⊢ (𝑟 = 𝑥 → (𝑟 × 𝑠) = (𝑥 × 𝑠))
14	13	eqeq2d 2205	. . . . . . . . 9 ⊢ (𝑟 = 𝑥 → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑠)))
15		xpeq2 4674	. . . . . . . . . 10 ⊢ (𝑠 = 𝑦 → (𝑥 × 𝑠) = (𝑥 × 𝑦))
16	15	eqeq2d 2205	. . . . . . . . 9 ⊢ (𝑠 = 𝑦 → (𝑤 = (𝑥 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑦)))
17	14, 16	cbvrex2v 2740	. . . . . . . 8 ⊢ (∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠) ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 𝑤 = (𝑥 × 𝑦))
18	12, 17	sylib 122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 𝑤 = (𝑥 × 𝑦))
19		simpr 110	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑤 = (𝑥 × 𝑦))
20		simplll 533	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝜑)
21		simplrl 535	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑥 ∈ 𝐽)
22		simplrr 536	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑦 ∈ 𝐾)
23	4	mopntopon 14611	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
24	2, 23	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
25	24	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐽 ∈ (TopOn‘𝑋))
26		simprl 529	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ∈ 𝐽)
27		toponss 14194	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
28	25, 26, 27	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ⊆ 𝑋)
29	5	mopntopon 14611	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
30	3, 29	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
31	30	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐾 ∈ (TopOn‘𝑌))
32		simprr 531	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
33		toponss 14194	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
34	31, 32, 33	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ 𝑌)
35		xpss12 4766	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
36	28, 34, 35	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
37	1, 2, 3	xmetxp 14675	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
38		unirnbl 14591	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → ∪ ran (ball‘𝑃) = (𝑋 × 𝑌))
39	37, 38	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∪ ran (ball‘𝑃) = (𝑋 × 𝑌))
40	39	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ∪ ran (ball‘𝑃) = (𝑋 × 𝑌))
41	36, 40	sseqtrrd 3218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 × 𝑦) ⊆ ∪ ran (ball‘𝑃))
42	2	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
43		simplrl 535	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥 ∈ 𝐽)
44		xp1st 6218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (𝑥 × 𝑦) → (1st ‘𝑗) ∈ 𝑥)
45	44	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st ‘𝑗) ∈ 𝑥)
46	4	mopni2 14651	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝐽 ∧ (1st ‘𝑗) ∈ 𝑥) → ∃𝑚 ∈ ℝ+ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
47	42, 43, 45, 46	syl3anc 1249	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑚 ∈ ℝ+ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
48	3	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
49		simplrr 536	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦 ∈ 𝐾)
50		xp2nd 6219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (𝑥 × 𝑦) → (2nd ‘𝑗) ∈ 𝑦)
51	50	adantl 277	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd ‘𝑗) ∈ 𝑦)
52	5	mopni2 14651	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑦 ∈ 𝐾 ∧ (2nd ‘𝑗) ∈ 𝑦) → ∃𝑛 ∈ ℝ+ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
53	48, 49, 51, 52	syl3anc 1249	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑛 ∈ ℝ+ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
54	53	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑛 ∈ ℝ+ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
55		blf 14578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃):((𝑋 × 𝑌) × ℝ*)⟶𝒫 (𝑋 × 𝑌))
56	37, 55	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ball‘𝑃):((𝑋 × 𝑌) × ℝ*)⟶𝒫 (𝑋 × 𝑌))
57	56	ffnd 5404	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
58	57	ad4antr 494	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*))
59	36	sselda 3179	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
60	59	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑗 ∈ (𝑋 × 𝑌))
61		rpxr 9727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℝ+ → 𝑚 ∈ ℝ*)
62	61	ad2antrl 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → 𝑚 ∈ ℝ*)
63	62	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ*)
64		rpxr 9727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℝ+ → 𝑛 ∈ ℝ*)
65	64	ad2antrl 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ*)
66		xrmincl 11409	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 ∈ ℝ* ∧ 𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*)
67	63, 65, 66	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*)
68		fnovrn 6066	. . . . . . . . . . . . . . . . . 18 ⊢ (((ball‘𝑃) Fn ((𝑋 × 𝑌) × ℝ*) ∧ 𝑗 ∈ (𝑋 × 𝑌) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ*) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∈ ran (ball‘𝑃))
69	58, 60, 67, 68	syl3anc 1249	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∈ ran (ball‘𝑃))
70		eleq2 2257	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑗 ∈ 𝑘 ↔ 𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))))
71		sseq1 3202	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑘 ⊆ (𝑥 × 𝑦) ↔ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
72	70, 71	anbi12d 473	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → ((𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)) ↔ (𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∧ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦))))
73	72	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) ∧ 𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))) → ((𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)) ↔ (𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∧ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦))))
74	37	ad4antr 494	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
75		simplrl 535	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑚 ∈ ℝ+)
76		simprl 529	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑛 ∈ ℝ+)
77		xrminrpcl 11417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+)
78	75, 76, 77	syl2anc 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+)
79		blcntr 14584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝑗 ∈ (𝑋 × 𝑌) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ+) → 𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )))
80	74, 60, 78, 79	syl3anc 1249	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )))
81	42	ad2antrr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑀 ∈ (∞Met‘𝑋))
82	48	ad2antrr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → 𝑁 ∈ (∞Met‘𝑌))
83	1, 81, 82, 67, 60	xmetxpbl 14676	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) = (((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) × ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < ))))
84	28	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥 ⊆ 𝑋)
85	84, 45	sseldd 3180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (1st ‘𝑗) ∈ 𝑋)
86	85	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (1st ‘𝑗) ∈ 𝑋)
87		xrmin1inf 11410	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ* ∧ 𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚)
88	63, 65, 87	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚)
89		ssbl 14594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ (1st ‘𝑗) ∈ 𝑋) ∧ (inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ* ∧ 𝑚 ∈ ℝ*) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑚) → ((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((1st ‘𝑗)(ball‘𝑀)𝑚))
90	81, 86, 67, 63, 88, 89	syl221anc 1260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((1st ‘𝑗)(ball‘𝑀)𝑚))
91		simplrr 536	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
92	90, 91	sstrd 3189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑥)
93	34	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦 ⊆ 𝑌)
94	93, 51	sseldd 3180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → (2nd ‘𝑗) ∈ 𝑌)
95	94	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (2nd ‘𝑗) ∈ 𝑌)
96		xrmin2inf 11411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ ℝ* ∧ 𝑛 ∈ ℝ*) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛)
97	63, 65, 96	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛)
98		ssbl 14594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd ‘𝑗) ∈ 𝑌) ∧ (inf({𝑚, 𝑛}, ℝ*, < ) ∈ ℝ* ∧ 𝑛 ∈ ℝ*) ∧ inf({𝑚, 𝑛}, ℝ*, < ) ≤ 𝑛) → ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((2nd ‘𝑗)(ball‘𝑁)𝑛))
99	82, 95, 67, 65, 97, 98	syl221anc 1260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ ((2nd ‘𝑗)(ball‘𝑁)𝑛))
100		simprr 531	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
101	99, 100	sstrd 3189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑦)
102		xpss12 4766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑥 ∧ ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ 𝑦) → (((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) × ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < ))) ⊆ (𝑥 × 𝑦))
103	92, 101, 102	syl2anc 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (((1st ‘𝑗)(ball‘𝑀)inf({𝑚, 𝑛}, ℝ*, < )) × ((2nd ‘𝑗)(ball‘𝑁)inf({𝑚, 𝑛}, ℝ*, < ))) ⊆ (𝑥 × 𝑦))
104	83, 103	eqsstrd 3215	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦))
105	80, 104	jca 306	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → (𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ∧ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) ⊆ (𝑥 × 𝑦)))
106	69, 73, 105	rspcedvd 2870	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) ∧ (𝑛 ∈ ℝ+ ∧ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))
107	54, 106	rexlimddv 2616	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) ∧ (𝑚 ∈ ℝ+ ∧ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))
108	47, 107	rexlimddv 2616	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))
109	108	ralrimiva 2567	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))
110	41, 109	jca 306	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝑥 × 𝑦) ⊆ ∪ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦))))
111		blex 14555	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → (ball‘𝑃) ∈ V)
112	37, 111	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ball‘𝑃) ∈ V)
113	112	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (ball‘𝑃) ∈ V)
114		rnexg 4927	. . . . . . . . . . . . 13 ⊢ ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
115		eltg2 14221	. . . . . . . . . . . . 13 ⊢ (ran (ball‘𝑃) ∈ V → ((𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)) ↔ ((𝑥 × 𝑦) ⊆ ∪ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))))
116	113, 114, 115	3syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)) ↔ ((𝑥 × 𝑦) ⊆ ∪ ran (ball‘𝑃) ∧ ∀𝑗 ∈ (𝑥 × 𝑦)∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))))
117	110, 116	mpbird 167	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
118	20, 21, 22, 117	syl12anc 1247	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → (𝑥 × 𝑦) ∈ (topGen‘ran (ball‘𝑃)))
119	19, 118	eqeltrd 2270	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑤 ∈ (topGen‘ran (ball‘𝑃)))
120	119	ex 115	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑤 = (𝑥 × 𝑦) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
121	120	rexlimdvva 2619	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) → (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 𝑤 = (𝑥 × 𝑦) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
122	18, 121	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) → 𝑤 ∈ (topGen‘ran (ball‘𝑃)))
123	122	ex 115	. . . . 5 ⊢ (𝜑 → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) → 𝑤 ∈ (topGen‘ran (ball‘𝑃))))
124	123	ssrdv 3185	. . . 4 ⊢ (𝜑 → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃)))
125	4	mopntop 14612	. . . . . . . 8 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
126	2, 125	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
127	5	mopntop 14612	. . . . . . . 8 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
128	3, 127	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
129		mpoexga 6265	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
130	126, 128, 129	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
131		rnexg 4927	. . . . . 6 ⊢ ((𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
132	130, 131	syl 14	. . . . 5 ⊢ (𝜑 → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V)
133	37, 111, 114	3syl 17	. . . . 5 ⊢ (𝜑 → ran (ball‘𝑃) ∈ V)
134		tgss3 14246	. . . . 5 ⊢ ((ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ∈ V ∧ ran (ball‘𝑃) ∈ V) → ((topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)) ↔ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃))))
135	132, 133, 134	syl2anc 411	. . . 4 ⊢ (𝜑 → ((topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)) ↔ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃))))
136	124, 135	mpbird 167	. . 3 ⊢ (𝜑 → (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ⊆ (topGen‘ran (ball‘𝑃)))
137		eqid 2193	. . . . 5 ⊢ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
138	137	txval 14423	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
139	126, 128, 138	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) = (topGen‘ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))))
140	6	mopnval 14610	. . . 4 ⊢ (𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) → 𝐿 = (topGen‘ran (ball‘𝑃)))
141	37, 140	syl 14	. . 3 ⊢ (𝜑 → 𝐿 = (topGen‘ran (ball‘𝑃)))
142	136, 139, 141	3sstr4d 3224	. 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) ⊆ 𝐿)
143	7, 142	eqssd 3196	1 ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ⊆ wss 3153  𝒫 cpw 3601  {cpr 3619  ∪ cuni 3835   class class class wbr 4029   × cxp 4657  ran crn 4660   Fn wfn 5249  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  1^st c1st 6191  2^nd c2nd 6192  supcsup 7041  infcinf 7042  ℝ^*cxr 8053   < clt 8054   ≤ cle 8055  ℝ⁺crp 9719  topGenctg 12865  ∞Metcxmet 14032  ballcbl 14034  MetOpencmopn 14037  Topctop 14165  TopOnctopon 14178   ×_t ctx 14420

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-xneg 9838  df-xadd 9839  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-topgen 12871  df-psmet 14039  df-xmet 14040  df-bl 14042  df-mopn 14043  df-top 14166  df-topon 14179  df-bases 14211  df-tx 14421

This theorem is referenced by:  txmetcnp  14686