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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({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
xmetxp.1 (𝜑𝑀 ∈ (∞Met‘𝑋))
xmetxp.2 (𝜑𝑁 ∈ (∞Met‘𝑌))
xmetxpbl.r (𝜑𝑅 ∈ ℝ*)
xmetxpbl.c (𝜑𝐶 ∈ (𝑋 × 𝑌))
Assertion
Ref Expression
xmetxpbl (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(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.
StepHypRef Expression
1 xmetxp.p . . . 4 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))
2 xmetxp.1 . . . 4 (𝜑𝑀 ∈ (∞Met‘𝑋))
3 xmetxp.2 . . . 4 (𝜑𝑁 ∈ (∞Met‘𝑌))
41, 2, 3xmetxp 12949 . . 3 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5 xmetxpbl.c . . 3 (𝜑𝐶 ∈ (𝑋 × 𝑌))
6 xmetxpbl.r . . 3 (𝜑𝑅 ∈ ℝ*)
7 blval 12831 . . 3 ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
84, 5, 6, 7syl3anc 1220 . 2 (𝜑 → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
95adantr 274 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ (𝑋 × 𝑌))
10 simpr 109 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
112adantr 274 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
12 xp1st 6114 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (1st𝐶) ∈ 𝑋)
139, 12syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝐶) ∈ 𝑋)
14 xp1st 6114 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (1st𝑡) ∈ 𝑋)
1514adantl 275 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝑡) ∈ 𝑋)
16 xmetcl 12794 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋 ∧ (1st𝑡) ∈ 𝑋) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
1711, 13, 15, 16syl3anc 1220 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
183adantr 274 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
19 xp2nd 6115 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (2nd𝐶) ∈ 𝑌)
209, 19syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝐶) ∈ 𝑌)
21 xp2nd 6115 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (2nd𝑡) ∈ 𝑌)
2221adantl 275 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝑡) ∈ 𝑌)
23 xmetcl 12794 . . . . . . . 8 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌 ∧ (2nd𝑡) ∈ 𝑌) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
2418, 20, 22, 23syl3anc 1220 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
25 xrmaxcl 11153 . . . . . . 7 ((((1st𝐶)𝑀(1st𝑡)) ∈ ℝ* ∧ ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*) → sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*)
2617, 24, 25syl2anc 409 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*)
27 fveq2 5469 . . . . . . . . . 10 (𝑢 = 𝐶 → (1st𝑢) = (1st𝐶))
28 fveq2 5469 . . . . . . . . . 10 (𝑣 = 𝑡 → (1st𝑣) = (1st𝑡))
2927, 28oveqan12d 5844 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝐶)𝑀(1st𝑡)))
30 fveq2 5469 . . . . . . . . . 10 (𝑢 = 𝐶 → (2nd𝑢) = (2nd𝐶))
31 fveq2 5469 . . . . . . . . . 10 (𝑣 = 𝑡 → (2nd𝑣) = (2nd𝑡))
3230, 31oveqan12d 5844 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝐶)𝑁(2nd𝑡)))
3329, 32preq12d 3645 . . . . . . . 8 ((𝑢 = 𝐶𝑣 = 𝑡) → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))})
3433supeq1d 6932 . . . . . . 7 ((𝑢 = 𝐶𝑣 = 𝑡) → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3534, 1ovmpoga 5951 . . . . . 6 ((𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑡 ∈ (𝑋 × 𝑌) ∧ sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*) → (𝐶𝑃𝑡) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
369, 10, 26, 35syl3anc 1220 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝐶𝑃𝑡) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3736breq1d 3976 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) < 𝑅))
386adantr 274 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑅 ∈ ℝ*)
39 xrmaxltsup 11159 . . . . 5 ((((1st𝐶)𝑀(1st𝑡)) ∈ ℝ* ∧ ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*𝑅 ∈ ℝ*) → (sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) < 𝑅 ↔ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)))
4017, 24, 38, 39syl3anc 1220 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) < 𝑅 ↔ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)))
4137, 40bitrd 187 . . 3 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)))
4241rabbidva 2700 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅} = {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)})
43 1st2nd2 6124 . . . . . . 7 (𝑛 ∈ (𝑋 × 𝑌) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
4443ad2antrl 482 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
45 xp1st 6114 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (1st𝑛) ∈ 𝑋)
4645ad2antrl 482 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ 𝑋)
47 simprrl 529 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → ((1st𝐶)𝑀(1st𝑛)) < 𝑅)
485, 12syl 14 . . . . . . . . 9 (𝜑 → (1st𝐶) ∈ 𝑋)
49 elbl 12833 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋𝑅 ∈ ℝ*) → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
502, 48, 6, 49syl3anc 1220 . . . . . . . 8 (𝜑 → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
5150adantr 274 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
5246, 47, 51mpbir2and 929 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
53 xp2nd 6115 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (2nd𝑛) ∈ 𝑌)
5453ad2antrl 482 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ 𝑌)
55 simprrr 530 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)
565, 19syl 14 . . . . . . . . 9 (𝜑 → (2nd𝐶) ∈ 𝑌)
57 elbl 12833 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌𝑅 ∈ ℝ*) → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
583, 56, 6, 57syl3anc 1220 . . . . . . . 8 (𝜑 → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
5958adantr 274 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
6054, 55, 59mpbir2and 929 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))
6144, 52, 60jca32 308 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))))
62 simprl 521 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
63 simprrl 529 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
6450adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
6563, 64mpbid 146 . . . . . . . . 9 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅))
6665simpld 111 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (1st𝑛) ∈ 𝑋)
67 simprrr 530 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))
6858adantr 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
6967, 68mpbid 146 . . . . . . . . 9 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))
7069simpld 111 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (2nd𝑛) ∈ 𝑌)
7162, 66, 70jca32 308 . . . . . . 7 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ 𝑋 ∧ (2nd𝑛) ∈ 𝑌)))
72 elxp6 6118 . . . . . . 7 (𝑛 ∈ (𝑋 × 𝑌) ↔ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ 𝑋 ∧ (2nd𝑛) ∈ 𝑌)))
7371, 72sylibr 133 . . . . . 6 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → 𝑛 ∈ (𝑋 × 𝑌))
7465simprd 113 . . . . . 6 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((1st𝐶)𝑀(1st𝑛)) < 𝑅)
7569simprd 113 . . . . . 6 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)
7673, 74, 75jca32 308 . . . . 5 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
7761, 76impbida 586 . . . 4 (𝜑 → ((𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)) ↔ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))))
78 fveq2 5469 . . . . . . . 8 (𝑡 = 𝑛 → (1st𝑡) = (1st𝑛))
7978oveq2d 5841 . . . . . . 7 (𝑡 = 𝑛 → ((1st𝐶)𝑀(1st𝑡)) = ((1st𝐶)𝑀(1st𝑛)))
8079breq1d 3976 . . . . . 6 (𝑡 = 𝑛 → (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ↔ ((1st𝐶)𝑀(1st𝑛)) < 𝑅))
81 fveq2 5469 . . . . . . . 8 (𝑡 = 𝑛 → (2nd𝑡) = (2nd𝑛))
8281oveq2d 5841 . . . . . . 7 (𝑡 = 𝑛 → ((2nd𝐶)𝑁(2nd𝑡)) = ((2nd𝐶)𝑁(2nd𝑛)))
8382breq1d 3976 . . . . . 6 (𝑡 = 𝑛 → (((2nd𝐶)𝑁(2nd𝑡)) < 𝑅 ↔ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))
8480, 83anbi12d 465 . . . . 5 (𝑡 = 𝑛 → ((((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅) ↔ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
8584elrab 2868 . . . 4 (𝑛 ∈ {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} ↔ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
86 elxp6 6118 . . . 4 (𝑛 ∈ (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅)) ↔ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))))
8777, 85, 863bitr4g 222 . . 3 (𝜑 → (𝑛 ∈ {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} ↔ 𝑛 ∈ (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅))))
8887eqrdv 2155 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅)))
898, 42, 883eqtrd 2194 1 (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(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  1st c1st 6087  2nd c2nd 6088  supcsup 6927  *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
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