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Theorem xmetxpbl 13675

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 13674	. . 3 ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5		xmetxpbl.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))
6		xmetxpbl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
7		blval 13556	. . 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 1238	. 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 6160	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (1^st ‘𝐶) ∈ 𝑋)
13	9, 12	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝐶) ∈ 𝑋)
14		xp1st 6160	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (1^st ‘𝑡) ∈ 𝑋)
15	14	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (1^st ‘𝑡) ∈ 𝑋)
16		xmetcl 13519	. . . . . . . 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 1238	. . . . . . 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 6161	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑋 × 𝑌) → (2^nd ‘𝐶) ∈ 𝑌)
20	9, 19	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝐶) ∈ 𝑌)
21		xp2nd 6161	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑡) ∈ 𝑌)
22	21	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → (2^nd ‘𝑡) ∈ 𝑌)
23		xmetcl 13519	. . . . . . . 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 1238	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑋 × 𝑌)) → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) ∈ ℝ^*)
25		xrmaxcl 11244	. . . . . . 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 5511	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (1^st ‘𝑢) = (1^st ‘𝐶))
28		fveq2 5511	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (1^st ‘𝑣) = (1^st ‘𝑡))
29	27, 28	oveqan12d 5888	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((1^st ‘𝑢)𝑀(1^st ‘𝑣)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑡)))
30		fveq2 5511	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (2^nd ‘𝑢) = (2^nd ‘𝐶))
31		fveq2 5511	. . . . . . . . . 10 ⊢ (𝑣 = 𝑡 → (2^nd ‘𝑣) = (2^nd ‘𝑡))
32	30, 31	oveqan12d 5888	. . . . . . . . 9 ⊢ ((𝑢 = 𝐶 ∧ 𝑣 = 𝑡) → ((2^nd ‘𝑢)𝑁(2^nd ‘𝑣)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)))
33	29, 32	preq12d 3676	. . . . . . . 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 6980	. . . . . . 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 5998	. . . . . 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 1238	. . . . 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 4010	. . . 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 11250	. . . . 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 1238	. . . 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 2725	. 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 6170	. . . . . . 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 6160	. . . . . . . 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 13558	. . . . . . . . 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 1238	. . . . . . . 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 944	. . . . . 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 6161	. . . . . . . 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 13558	. . . . . . . . 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 1238	. . . . . . . 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 944	. . . . . 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 6164	. . . . . . 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 5511	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (1^st ‘𝑡) = (1^st ‘𝑛))
79	78	oveq2d 5885	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((1^st ‘𝐶)𝑀(1^st ‘𝑡)) = ((1^st ‘𝐶)𝑀(1^st ‘𝑛)))
80	79	breq1d 4010	. . . . . 6 ⊢ (𝑡 = 𝑛 → (((1^st ‘𝐶)𝑀(1^st ‘𝑡)) < 𝑅 ↔ ((1^st ‘𝐶)𝑀(1^st ‘𝑛)) < 𝑅))
81		fveq2 5511	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (2^nd ‘𝑡) = (2^nd ‘𝑛))
82	81	oveq2d 5885	. . . . . . 7 ⊢ (𝑡 = 𝑛 → ((2^nd ‘𝐶)𝑁(2^nd ‘𝑡)) = ((2^nd ‘𝐶)𝑁(2^nd ‘𝑛)))
83	82	breq1d 4010	. . . . . 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 2893	. . . 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 6164	. . . 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 2175	. 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 2214	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 1353 ∈ wcel 2148 {crab 2459 {cpr 3592 ⟨cop 3594 class class class wbr 4000 × cxp 4621 ‘cfv 5212 (class class class)co 5869 ∈ cmpo 5871 1^st c1st 6133 2^nd c2nd 6134 supcsup 6975 ℝ^*cxr 7981 < clt 7982 ∞Metcxmet 13147 ballcbl 13149

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-map 6644 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-xneg 9759 df-xadd 9760 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-topgen 12657 df-psmet 13154 df-xmet 13155 df-bl 13157 df-mopn 13158 df-top 13163 df-topon 13176 df-bases 13208

This theorem is referenced by: xmettxlem 13676 xmettx 13677

W3C validator