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Theorem xmetxpbl 14898

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 14897	. . 3 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		xmetxpbl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))
6		xmetxpbl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		blval 14779	. . 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 6241	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (1^st ‘𝐶) ∈ 𝑋)
13	9, 12	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝐶) ∈ 𝑋)
14		xp1st 6241	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝑡) ∈ 𝑋)
16		xmetcl 14742	. . . . . . . 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 6242	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (2^nd ‘𝐶) ∈ 𝑌)
20	9, 19	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝐶) ∈ 𝑌)
21		xp2nd 6242	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
22	21	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝑡) ∈ 𝑌)
23		xmetcl 14742	. . . . . . . 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 11482	. . . . . . 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 5570	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (1^st ‘𝑢) = (1^st ‘𝐶))
28		fveq2 5570	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (1^st ‘𝑣) = (1^st ‘𝑡))
29	27, 28	oveqan12d 5953	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑡)))
30		fveq2 5570	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (2^nd ‘𝑢) = (2^nd ‘𝐶))
31		fveq2 5570	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (2^nd ‘𝑣) = (2^nd ‘𝑡))
32	30, 31	oveqan12d 5953	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)))
33	29, 32	preq12d 3717	. . . . . . . 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 7071	. . . . . . 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 6065	. . . . . 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 4053	. . . 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 11488	. . . . 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 2759	. 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 6251	. . . . . . 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 6241	. . . . . . . 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 14781	. . . . . . . . 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 6242	. . . . . . . 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 14781	. . . . . . . . 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 6245	. . . . . . 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 5570	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (1^st ‘𝑡) = (1^st ‘𝑛))
79	78	oveq2d 5950	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑛)))
80	79	breq1d 4053	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ↔ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅))
81		fveq2 5570	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (2^nd ‘𝑡) = (2^nd ‘𝑛))
82	81	oveq2d 5950	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)))
83	82	breq1d 4053	. . . . . 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 2928	. . . 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 6245	. . . 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 2202	. 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 2241	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 1372 ∈ wcel 2175 {crab 2487 {cpr 3633 ⟨cop 3635 class class class wbr 4043 × cxp 4671 ‘cfv 5268 (class class class)co 5934 ∈ cmpo 5936 1^st c1st 6214 2^nd c2nd 6215 supcsup 7066 ℝ^*cxr 8088 < clt 8089 ∞Metcxmet 14216 ballcbl 14218

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-iinf 4634 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 ax-arch 8026 ax-caucvg 8027

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4338 df-po 4341 df-iso 4342 df-iord 4411 df-on 4413 df-ilim 4414 df-suc 4416 df-iom 4637 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-res 4685 df-ima 4686 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-f1 5273 df-fo 5274 df-f1o 5275 df-fv 5276 df-isom 5277 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-1st 6216 df-2nd 6217 df-recs 6381 df-frec 6467 df-map 6727 df-sup 7068 df-inf 7069 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 df-inn 9019 df-2 9077 df-3 9078 df-4 9079 df-n0 9278 df-z 9355 df-uz 9631 df-q 9723 df-rp 9758 df-xneg 9876 df-xadd 9877 df-seqfrec 10574 df-exp 10665 df-cj 11072 df-re 11073 df-im 11074 df-rsqrt 11228 df-abs 11229 df-topgen 13010 df-psmet 14223 df-xmet 14224 df-bl 14226 df-mopn 14227 df-top 14388 df-topon 14401 df-bases 14433

This theorem is referenced by: xmettxlem 14899 xmettx 14900

W3C validator