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Theorem xmetxpbl 14676

Description: The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point 𝐶 with radius 𝑅. (Contributed by Jim Kingdon, 22-Oct-2023.)

**Hypotheses**
Ref	Expression
xmetxp.p	⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^*, < ))
xmetxp.1	⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
xmetxp.2	⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
xmetxpbl.r	⊢ (𝜑 → 𝑅 ∈ ℝ^*)
xmetxpbl.c	⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))

**Assertion**
Ref	Expression
xmetxpbl	⊢ (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1^st ‘𝐶)(ball‘𝑀)𝑅) × ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))

Distinct variable groups: 𝑢,𝐶,𝑣 𝑢,𝑀,𝑣 𝑢,𝑁,𝑣 𝑢,𝑋,𝑣 𝑢,𝑌,𝑣 Allowed substitution hints: 𝜑(𝑣,𝑢) 𝑃(𝑣,𝑢) 𝑅(𝑣,𝑢)

Proof of Theorem xmetxpbl Dummy variables 𝑛 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetxp.p	. . . 4 ⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^*, < ))
2		xmetxp.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))
3		xmetxp.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))
4	1, 2, 3	xmetxp 14675	. . 3 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		xmetxpbl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))
6		xmetxpbl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		blval 14557	. . 3 ⊢ ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑅 ∈ ℝ^*) → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
8	4, 5, 6, 7	syl3anc 1249	. 2 ⊢ (𝜑 → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
9	5	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ (𝑋 × 𝑌))
10		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
11	2	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
12		xp1st 6218	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (1^st ‘𝐶) ∈ 𝑋)
13	9, 12	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝐶) ∈ 𝑋)
14		xp1st 6218	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝑡) ∈ 𝑋)
16		xmetcl 14520	. . . . . . . 8 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝐶) ∈ 𝑋 ∧ (1^st ‘𝑡) ∈ 𝑋) → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) ∈ ℝ^*)
17	11, 13, 15, 16	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) ∈ ℝ^*)
18	3	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
19		xp2nd 6219	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (2^nd ‘𝐶) ∈ 𝑌)
20	9, 19	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝐶) ∈ 𝑌)
21		xp2nd 6219	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
22	21	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝑡) ∈ 𝑌)
23		xmetcl 14520	. . . . . . . 8 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝐶) ∈ 𝑌 ∧ (2^nd ‘𝑡) ∈ 𝑌) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^*)
24	18, 20, 22, 23	syl3anc 1249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^*)
25		xrmaxcl 11395	. . . . . . 7 ⊢ ((((1^st ‘𝐶)𝑀(1^st ‘𝑡)) ∈ ℝ^* ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^) → sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^, < ) ∈ ℝ^*)
26	17, 24, 25	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^, < ) ∈ ℝ^)
27		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (1^st ‘𝑢) = (1^st ‘𝐶))
28		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (1^st ‘𝑣) = (1^st ‘𝑡))
29	27, 28	oveqan12d 5937	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑡)))
30		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (2^nd ‘𝑢) = (2^nd ‘𝐶))
31		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (2^nd ‘𝑣) = (2^nd ‘𝑡))
32	30, 31	oveqan12d 5937	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)))
33	29, 32	preq12d 3703	. . . . . . . 8 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → {((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))} = {((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))})
34	33	supeq1d 7046	. . . . . . 7 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → sup({((1^st ‘𝑢)𝑀(1^st ‘𝑣)), ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣))}, ℝ^, < ) = sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^, < ))
35	34, 1	ovmpoga 6048	. . . . . 6 ⊢ ((𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌) ∧ sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^, < ) ∈ ℝ^) → (𝐶𝑃𝑡) = sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^*, < ))
36	9, 10, 26, 35	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (𝐶𝑃𝑡) = sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^*, < ))
37	36	breq1d 4039	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^*, < ) < 𝑅))
38	6	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑅 ∈ ℝ^*)
39		xrmaxltsup 11401	. . . . 5 ⊢ ((((1^st ‘𝐶)𝑀(1^st ‘𝑡)) ∈ ℝ^* ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → (sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^, < ) < 𝑅 ↔ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)))
40	17, 24, 38, 39	syl3anc 1249	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^*, < ) < 𝑅 ↔ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)))
41	37, 40	bitrd 188	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)))
42	41	rabbidva 2748	. 2 ⊢ (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅} = {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)})
43		1st2nd2 6228	. . . . . . 7 ⊢ (𝑛 ∈ (𝑋 × 𝑌) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
44	43	ad2antrl 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
45		xp1st 6218	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑋 × 𝑌) → (1^st ‘𝑛) ∈ 𝑋)
46	45	ad2antrl 490	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → (1^st ‘𝑛) ∈ 𝑋)
47		simprrl 539	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅)
48	5, 12	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐶) ∈ 𝑋)
49		elbl 14559	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (1^st ‘𝐶) ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ↔ ((1^st ‘𝑛) ∈ 𝑋 ∧ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅)))
50	2, 48, 6, 49	syl3anc 1249	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ↔ ((1^st ‘𝑛) ∈ 𝑋 ∧ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅)))
51	50	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ↔ ((1^st ‘𝑛) ∈ 𝑋 ∧ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅)))
52	46, 47, 51	mpbir2and 946	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → (1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅))
53		xp2nd 6219	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑛) ∈ 𝑌)
54	53	ad2antrl 490	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → (2^nd ‘𝑛) ∈ 𝑌)
55		simprrr 540	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)
56	5, 19	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝐶) ∈ 𝑌)
57		elbl 14559	. . . . . . . . 9 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (2^nd ‘𝐶) ∈ 𝑌 ∧ 𝑅 ∈ ℝ^*) → ((2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅) ↔ ((2^nd ‘𝑛) ∈ 𝑌 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
58	3, 56, 6, 57	syl3anc 1249	. . . . . . . 8 ⊢ (𝜑 → ((2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅) ↔ ((2^nd ‘𝑛) ∈ 𝑌 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
59	58	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → ((2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅) ↔ ((2^nd ‘𝑛) ∈ 𝑌 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
60	54, 55, 59	mpbir2and 946	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅))
61	44, 52, 60	jca32 310	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))) → (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅))))
62		simprl 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
63		simprrl 539	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → (1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅))
64	50	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ↔ ((1^st ‘𝑛) ∈ 𝑋 ∧ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅)))
65	63, 64	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → ((1^st ‘𝑛) ∈ 𝑋 ∧ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅))
66	65	simpld 112	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → (1^st ‘𝑛) ∈ 𝑋)
67		simprrr 540	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅))
68	58	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → ((2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅) ↔ ((2^nd ‘𝑛) ∈ 𝑌 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
69	67, 68	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → ((2^nd ‘𝑛) ∈ 𝑌 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))
70	69	simpld 112	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → (2^nd ‘𝑛) ∈ 𝑌)
71	62, 66, 70	jca32 310	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ 𝑋 ∧ (2^nd ‘𝑛) ∈ 𝑌)))
72		elxp6 6222	. . . . . . 7 ⊢ (𝑛 ∈ (𝑋 × 𝑌) ↔ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ 𝑋 ∧ (2^nd ‘𝑛) ∈ 𝑌)))
73	71, 72	sylibr 134	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → 𝑛 ∈ (𝑋 × 𝑌))
74	65	simprd 114	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅)
75	69	simprd 114	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)
76	73, 74, 75	jca32 310	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))) → (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
77	61, 76	impbida 596	. . . 4 ⊢ (𝜑 → ((𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)) ↔ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))))
78		fveq2 5554	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (1^st ‘𝑡) = (1^st ‘𝑛))
79	78	oveq2d 5934	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑛)))
80	79	breq1d 4039	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ↔ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅))
81		fveq2 5554	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (2^nd ‘𝑡) = (2^nd ‘𝑛))
82	81	oveq2d 5934	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)))
83	82	breq1d 4039	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅 ↔ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))
84	80, 83	anbi12d 473	. . . . 5 ⊢ (𝑡 = 𝑛 → ((((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅) ↔ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
85	84	elrab 2916	. . . 4 ⊢ (𝑛 ∈ {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)} ↔ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅)))
86		elxp6 6222	. . . 4 ⊢ (𝑛 ∈ (((1^st ‘𝐶)(ball‘𝑀)𝑅) × ((2^nd ‘𝐶)(ball‘𝑁)𝑅)) ↔ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ ((1^st ‘𝐶)(ball‘𝑀)𝑅) ∧ (2^nd ‘𝑛) ∈ ((2^nd ‘𝐶)(ball‘𝑁)𝑅))))
87	77, 85, 86	3bitr4g 223	. . 3 ⊢ (𝜑 → (𝑛 ∈ {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)} ↔ 𝑛 ∈ (((1^st ‘𝐶)(ball‘𝑀)𝑅) × ((2^nd ‘𝐶)(ball‘𝑁)𝑅))))
88	87	eqrdv 2191	. 2 ⊢ (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ∧ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅)} = (((1^st ‘𝐶)(ball‘𝑀)𝑅) × ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))
89	8, 42, 88	3eqtrd 2230	1 ⊢ (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1^st ‘𝐶)(ball‘𝑀)𝑅) × ((2^nd ‘𝐶)(ball‘𝑁)𝑅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 {crab 2476 {cpr 3619 ⟨cop 3621 class class class wbr 4029 × cxp 4657 ‘cfv 5254 (class class class)co 5918 ∈ cmpo 5920 1^st c1st 6191 2^nd c2nd 6192 supcsup 7041 ℝ^*cxr 8053 < clt 8054 ∞Metcxmet 14032 ballcbl 14034

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

This theorem is referenced by: xmettxlem 14677 xmettx 14678

W3C validator