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Theorem xmetxpbl 15231

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 15230	. . 3 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		xmetxpbl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))
6		xmetxpbl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		blval 15112	. . 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 1273	. 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 6327	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (1^st ‘𝐶) ∈ 𝑋)
13	9, 12	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝐶) ∈ 𝑋)
14		xp1st 6327	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝑡) ∈ 𝑋)
16		xmetcl 15075	. . . . . . . 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 1273	. . . . . . 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 6328	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (2^nd ‘𝐶) ∈ 𝑌)
20	9, 19	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝐶) ∈ 𝑌)
21		xp2nd 6328	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
22	21	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝑡) ∈ 𝑌)
23		xmetcl 15075	. . . . . . . 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 1273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^*)
25		xrmaxcl 11812	. . . . . . 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 5639	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (1^st ‘𝑢) = (1^st ‘𝐶))
28		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (1^st ‘𝑣) = (1^st ‘𝑡))
29	27, 28	oveqan12d 6036	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑡)))
30		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (2^nd ‘𝑢) = (2^nd ‘𝐶))
31		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (2^nd ‘𝑣) = (2^nd ‘𝑡))
32	30, 31	oveqan12d 6036	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)))
33	29, 32	preq12d 3756	. . . . . . . 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 7185	. . . . . . 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 6150	. . . . . 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 1273	. . . . 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 4098	. . . 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 11818	. . . . 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 1273	. . . 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 2790	. 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 6337	. . . . . . 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 6327	. . . . . . . 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 541	. . . . . . 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 15114	. . . . . . . . 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 1273	. . . . . . . 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 952	. . . . . 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 6328	. . . . . . . 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 542	. . . . . . 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 15114	. . . . . . . . 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 1273	. . . . . . . 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 952	. . . . . 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 531	. . . . . . . 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 541	. . . . . . . . . 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 542	. . . . . . . . . 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 6331	. . . . . . 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 600	. . . 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 5639	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (1^st ‘𝑡) = (1^st ‘𝑛))
79	78	oveq2d 6033	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑛)))
80	79	breq1d 4098	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ↔ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅))
81		fveq2 5639	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (2^nd ‘𝑡) = (2^nd ‘𝑛))
82	81	oveq2d 6033	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)))
83	82	breq1d 4098	. . . . . 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 2962	. . . 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 6331	. . . 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 2229	. 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 2268	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 1397 ∈ wcel 2202 {crab 2514 {cpr 3670 ⟨cop 3672 class class class wbr 4088 × cxp 4723 ‘cfv 5326 (class class class)co 6017 ∈ cmpo 6019 1^st c1st 6300 2^nd c2nd 6301 supcsup 7180 ℝ^*cxr 8212 < clt 8213 ∞Metcxmet 14549 ballcbl 14551

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-map 6818 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-bl 14559 df-mopn 14560 df-top 14721 df-topon 14734 df-bases 14766

This theorem is referenced by: xmettxlem 15232 xmettx 15233

W3C validator