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Theorem xmetxpbl 13302
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 13301 . . 3 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5 xmetxpbl.c . . 3 (𝜑𝐶 ∈ (𝑋 × 𝑌))
6 xmetxpbl.r . . 3 (𝜑𝑅 ∈ ℝ*)
7 blval 13183 . . 3 ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
84, 5, 6, 7syl3anc 1233 . 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 6144 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (1st𝐶) ∈ 𝑋)
139, 12syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝐶) ∈ 𝑋)
14 xp1st 6144 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (1st𝑡) ∈ 𝑋)
1514adantl 275 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝑡) ∈ 𝑋)
16 xmetcl 13146 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋 ∧ (1st𝑡) ∈ 𝑋) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
1711, 13, 15, 16syl3anc 1233 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
183adantr 274 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
19 xp2nd 6145 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (2nd𝐶) ∈ 𝑌)
209, 19syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝐶) ∈ 𝑌)
21 xp2nd 6145 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (2nd𝑡) ∈ 𝑌)
2221adantl 275 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝑡) ∈ 𝑌)
23 xmetcl 13146 . . . . . . . 8 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌 ∧ (2nd𝑡) ∈ 𝑌) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
2418, 20, 22, 23syl3anc 1233 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
25 xrmaxcl 11215 . . . . . . 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 5496 . . . . . . . . . 10 (𝑢 = 𝐶 → (1st𝑢) = (1st𝐶))
28 fveq2 5496 . . . . . . . . . 10 (𝑣 = 𝑡 → (1st𝑣) = (1st𝑡))
2927, 28oveqan12d 5872 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝐶)𝑀(1st𝑡)))
30 fveq2 5496 . . . . . . . . . 10 (𝑢 = 𝐶 → (2nd𝑢) = (2nd𝐶))
31 fveq2 5496 . . . . . . . . . 10 (𝑣 = 𝑡 → (2nd𝑣) = (2nd𝑡))
3230, 31oveqan12d 5872 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝐶)𝑁(2nd𝑡)))
3329, 32preq12d 3668 . . . . . . . 8 ((𝑢 = 𝐶𝑣 = 𝑡) → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))})
3433supeq1d 6964 . . . . . . 7 ((𝑢 = 𝐶𝑣 = 𝑡) → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3534, 1ovmpoga 5982 . . . . . 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 1233 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝐶𝑃𝑡) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3736breq1d 3999 . . . 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 11221 . . . . 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 1233 . . . 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 2718 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅} = {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)})
43 1st2nd2 6154 . . . . . . 7 (𝑛 ∈ (𝑋 × 𝑌) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
4443ad2antrl 487 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
45 xp1st 6144 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (1st𝑛) ∈ 𝑋)
4645ad2antrl 487 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ 𝑋)
47 simprrl 534 . . . . . . 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 13185 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋𝑅 ∈ ℝ*) → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
502, 48, 6, 49syl3anc 1233 . . . . . . . 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 939 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
53 xp2nd 6145 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (2nd𝑛) ∈ 𝑌)
5453ad2antrl 487 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ 𝑌)
55 simprrr 535 . . . . . . 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 13185 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌𝑅 ∈ ℝ*) → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
583, 56, 6, 57syl3anc 1233 . . . . . . . 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 939 . . . . . 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 526 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
63 simprrl 534 . . . . . . . . . 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 535 . . . . . . . . . 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 6148 . . . . . . 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 591 . . . 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 5496 . . . . . . . 8 (𝑡 = 𝑛 → (1st𝑡) = (1st𝑛))
7978oveq2d 5869 . . . . . . 7 (𝑡 = 𝑛 → ((1st𝐶)𝑀(1st𝑡)) = ((1st𝐶)𝑀(1st𝑛)))
8079breq1d 3999 . . . . . 6 (𝑡 = 𝑛 → (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ↔ ((1st𝐶)𝑀(1st𝑛)) < 𝑅))
81 fveq2 5496 . . . . . . . 8 (𝑡 = 𝑛 → (2nd𝑡) = (2nd𝑛))
8281oveq2d 5869 . . . . . . 7 (𝑡 = 𝑛 → ((2nd𝐶)𝑁(2nd𝑡)) = ((2nd𝐶)𝑁(2nd𝑛)))
8382breq1d 3999 . . . . . 6 (𝑡 = 𝑛 → (((2nd𝐶)𝑁(2nd𝑡)) < 𝑅 ↔ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))
8480, 83anbi12d 470 . . . . 5 (𝑡 = 𝑛 → ((((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅) ↔ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
8584elrab 2886 . . . 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 6148 . . . 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 2168 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅)))
898, 42, 883eqtrd 2207 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 1348  wcel 2141  {crab 2452  {cpr 3584  cop 3586   class class class wbr 3989   × cxp 4609  cfv 5198  (class class class)co 5853  cmpo 5855  1st c1st 6117  2nd c2nd 6118  supcsup 6959  *cxr 7953   < clt 7954  ∞Metcxmet 12774  ballcbl 12776
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835
This theorem is referenced by:  xmettxlem  13303  xmettx  13304
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