Theorem xmettx 15501

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 15500	. 2 ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))
8		eqid 2234	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
9	8	elrnmpog 6174	. . . . . . . . . . 11 ⊢ (𝑤 ∈ V → (𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ↔ ∃𝑟 ∈ 𝐽 ∃𝑠 ∈ 𝐾 𝑤 = (𝑟 × 𝑠)))
10	9	elv 2819	. . . . . . . . . 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 4768	. . . . . . . . . 10 ⊢ (𝑟 = 𝑥 → (𝑟 × 𝑠) = (𝑥 × 𝑠))
14	13	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑟 = 𝑥 → (𝑤 = (𝑟 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑠)))
15		xpeq2 4769	. . . . . . . . . 10 ⊢ (𝑠 = 𝑦 → (𝑥 × 𝑠) = (𝑥 × 𝑦))
16	15	eqeq2d 2246	. . . . . . . . 9 ⊢ (𝑠 = 𝑦 → (𝑤 = (𝑥 × 𝑠) ↔ 𝑤 = (𝑥 × 𝑦)))
17	14, 16	cbvrex2v 2794	. . . . . . . 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 535	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝜑)
21		simplrl 537	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑥 ∈ 𝐽)
22		simplrr 538	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑤 ∈ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑤 = (𝑥 × 𝑦)) → 𝑦 ∈ 𝐾)
23	4	mopntopon 15434	. . . . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ∈ 𝐽)
27		toponss 15017	. . . . . . . . . . . . . . . 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 15434	. . . . . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ∈ 𝐾)
33		toponss 15017	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑦 ∈ 𝐾) → 𝑦 ⊆ 𝑌)
34	31, 32, 33	syl2anc 411	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ 𝑌)
35		xpss12 4862	. . . . . . . . . . . . . . 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 15498	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
38		unirnbl 15414	. . . . . . . . . . . . . . . 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 3281	. . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑥 ∈ 𝐽)
44		xp1st 6372	. . . . . . . . . . . . . . . . 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 15474	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝐽 ∧ (1st ‘𝑗) ∈ 𝑥) → ∃𝑚 ∈ ℝ+ ((1st ‘𝑗)(ball‘𝑀)𝑚) ⊆ 𝑥)
47	42, 43, 45, 46	syl3anc 1274	. . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → 𝑦 ∈ 𝐾)
50		xp2nd 6373	. . . . . . . . . . . . . . . . . . 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 15474	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑦 ∈ 𝐾 ∧ (2nd ‘𝑗) ∈ 𝑦) → ∃𝑛 ∈ ℝ+ ((2nd ‘𝑗)(ball‘𝑁)𝑛) ⊆ 𝑦)
53	48, 49, 51, 52	syl3anc 1274	. . . . . . . . . . . . . . . . 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 15401	. . . . . . . . . . . . . . . . . . . . 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 5514	. . . . . . . . . . . . . . . . . . 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 3242	. . . . . . . . . . . . . . . . . . 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 10012	. . . . . . . . . . . . . . . . . . . . 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 10012	. . . . . . . . . . . . . . . . . . . 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 11976	. . . . . . . . . . . . . . . . . . 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 6210	. . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . 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 2298	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < )) → (𝑗 ∈ 𝑘 ↔ 𝑗 ∈ (𝑗(ball‘𝑃)inf({𝑚, 𝑛}, ℝ*, < ))))
71		sseq1 3265	. . . . . . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . . . . . 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 11984	. . . . . . . . . . . . . . . . . . . 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 15407	. . . . . . . . . . . . . . . . . . 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 1274	. . . . . . . . . . . . . . . . . 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 15499	. . . . . . . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . . . . . . . . . . . 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 11977	. . . . . . . . . . . . . . . . . . . . . . 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 15417	. . . . . . . . . . . . . . . . . . . . . 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 1285	. . . . . . . . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . . . . . 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 3252	. . . . . . . . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . . . . . . . . . . . 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 11978	. . . . . . . . . . . . . . . . . . . . . . 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 15417	. . . . . . . . . . . . . . . . . . . . . 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 1285	. . . . . . . . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . . . . . . . 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 3252	. . . . . . . . . . . . . . . . . . . 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 4862	. . . . . . . . . . . . . . . . . . . 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 3278	. . . . . . . . . . . . . . . . . 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 2929	. . . . . . . . . . . . . . . 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 2667	. . . . . . . . . . . . . . 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 2667	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) ∧ 𝑗 ∈ (𝑥 × 𝑦)) → ∃𝑘 ∈ ran (ball‘𝑃)(𝑗 ∈ 𝑘 ∧ 𝑘 ⊆ (𝑥 × 𝑦)))
109	108	ralrimiva 2617	. . . . . . . . . . . . 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 15378	. . . . . . . . . . . . . . 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 5027	. . . . . . . . . . . . 13 ⊢ ((ball‘𝑃) ∈ V → ran (ball‘𝑃) ∈ V)
115		eltg2 15044	. . . . . . . . . . . . 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 1272	. . . . . . . . . 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 2311	. . . . . . . . 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 2670	. . . . . . 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 3248	. . . 4 ⊢ (𝜑 → ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) ⊆ (topGen‘ran (ball‘𝑃)))
125	4	mopntop 15435	. . . . . . . 8 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
126	2, 125	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
127	5	mopntop 15435	. . . . . . . 8 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ Top)
128	3, 127	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
129		mpoexga 6421	. . . . . . 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 5027	. . . . . 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 15069	. . . . 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 2234	. . . . 5 ⊢ ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠)) = ran (𝑟 ∈ 𝐽, 𝑠 ∈ 𝐾 ↦ (𝑟 × 𝑠))
138	137	txval 15246	. . . 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 15433	. . . 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 3287	. 2 ⊢ (𝜑 → (𝐽 ×t 𝐾) ⊆ 𝐿)
143	7, 142	eqssd 3259	1 ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  Vcvv 2815   ⊆ wss 3214  𝒫 cpw 3674  {cpr 3695  ∪ cuni 3919   class class class wbr 4114   × cxp 4752  ran crn 4755   Fn wfn 5352  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060  1^st c1st 6345  2^nd c2nd 6346  supcsup 7286  infcinf 7287  ℝ^*cxr 8323   < clt 8324   ≤ cle 8325  ℝ⁺crp 10004  topGenctg 13551  ∞Metcxmet 14810  ballcbl 14812  MetOpencmopn 14815  Topctop 14988  TopOnctopon 15001   ×_t ctx 15243

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-tx 15244

This theorem is referenced by:  txmetcnp  15509