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Theorem xmetxpbl 12950

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 12949	. . 3 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		xmetxpbl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))
6		xmetxpbl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		blval 12831	. . 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 1220	. 2 ⊢ (𝜑 → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
9	5	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ (𝑋 × 𝑌))
10		simpr 109	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
11	2	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
12		xp1st 6114	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (1^st ‘𝐶) ∈ 𝑋)
13	9, 12	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝐶) ∈ 𝑋)
14		xp1st 6114	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
15	14	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝑡) ∈ 𝑋)
16		xmetcl 12794	. . . . . . . 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 1220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) ∈ ℝ^*)
18	3	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
19		xp2nd 6115	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (2^nd ‘𝐶) ∈ 𝑌)
20	9, 19	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝐶) ∈ 𝑌)
21		xp2nd 6115	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
22	21	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝑡) ∈ 𝑌)
23		xmetcl 12794	. . . . . . . 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 1220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^*)
25		xrmaxcl 11153	. . . . . . 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 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → sup({((1^st ‘𝐶)𝑀(1^st ‘𝑡)), ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡))}, ℝ^, < ) ∈ ℝ^)
27		fveq2 5469	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (1^st ‘𝑢) = (1^st ‘𝐶))
28		fveq2 5469	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (1^st ‘𝑣) = (1^st ‘𝑡))
29	27, 28	oveqan12d 5844	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑡)))
30		fveq2 5469	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (2^nd ‘𝑢) = (2^nd ‘𝐶))
31		fveq2 5469	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (2^nd ‘𝑣) = (2^nd ‘𝑡))
32	30, 31	oveqan12d 5844	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)))
33	29, 32	preq12d 3645	. . . . . . . 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 6932	. . . . . . 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 5951	. . . . . 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 1220	. . . . 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 3976	. . . 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 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → 𝑅 ∈ ℝ^*)
39		xrmaxltsup 11159	. . . . 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 1220	. . . 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 187	. . 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 2700	. 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 6124	. . . . . . 7 ⊢ (𝑛 ∈ (𝑋 × 𝑌) → 𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩)
44	43	ad2antrl 482	. . . . . 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 6114	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑋 × 𝑌) → (1^st ‘𝑛) ∈ 𝑋)
46	45	ad2antrl 482	. . . . . . 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 529	. . . . . . 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 12833	. . . . . . . . 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 1220	. . . . . . . 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 274	. . . . . . 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 929	. . . . . 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 6115	. . . . . . . 8 ⊢ (𝑛 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑛) ∈ 𝑌)
54	53	ad2antrl 482	. . . . . . 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 530	. . . . . . 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 12833	. . . . . . . . 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 1220	. . . . . . . 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 274	. . . . . . 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 929	. . . . . 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 308	. . . . 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 521	. . . . . . . 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 529	. . . . . . . . . 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 274	. . . . . . . . . 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 146	. . . . . . . . 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 111	. . . . . . . 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 530	. . . . . . . . . 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 274	. . . . . . . . . 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 146	. . . . . . . . 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 111	. . . . . . . 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 308	. . . . . . 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 6118	. . . . . . 7 ⊢ (𝑛 ∈ (𝑋 × 𝑌) ↔ (𝑛 = ⟨(1^st ‘𝑛), (2^nd ‘𝑛)⟩ ∧ ((1^st ‘𝑛) ∈ 𝑋 ∧ (2^nd ‘𝑛) ∈ 𝑌)))
73	71, 72	sylibr 133	. . . . . 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 113	. . . . . 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 113	. . . . . 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 308	. . . . 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 586	. . . 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 5469	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (1^st ‘𝑡) = (1^st ‘𝑛))
79	78	oveq2d 5841	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑛)))
80	79	breq1d 3976	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ↔ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅))
81		fveq2 5469	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (2^nd ‘𝑡) = (2^nd ‘𝑛))
82	81	oveq2d 5841	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)))
83	82	breq1d 3976	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) < 𝑅 ↔ ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)) < 𝑅))
84	80, 83	anbi12d 465	. . . . 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 2868	. . . 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 6118	. . . 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 222	. . 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 2155	. 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 2194	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 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 {crab 2439 {cpr 3561 ⟨cop 3563 class class class wbr 3966 × cxp 4585 ‘cfv 5171 (class class class)co 5825 ∈ cmpo 5827 1^st c1st 6087 2^nd c2nd 6088 supcsup 6927 ℝ^*cxr 7912 < clt 7913 ∞Metcxmet 12422 ballcbl 12424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-map 6596 df-sup 6929 df-inf 6930 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-xneg 9680 df-xadd 9681 df-seqfrec 10349 df-exp 10423 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-topgen 12414 df-psmet 12429 df-xmet 12430 df-bl 12432 df-mopn 12433 df-top 12438 df-topon 12451 df-bases 12483

This theorem is referenced by: xmettxlem 12951 xmettx 12952

W3C validator