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Theorem xmetxpbl 12572
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 12571 . . 3 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5 xmetxpbl.c . . 3 (𝜑𝐶 ∈ (𝑋 × 𝑌))
6 xmetxpbl.r . . 3 (𝜑𝑅 ∈ ℝ*)
7 blval 12453 . . 3 ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
84, 5, 6, 7syl3anc 1199 . 2 (𝜑 → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
95adantr 272 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ (𝑋 × 𝑌))
10 simpr 109 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
112adantr 272 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
12 xp1st 6029 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (1st𝐶) ∈ 𝑋)
139, 12syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝐶) ∈ 𝑋)
14 xp1st 6029 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (1st𝑡) ∈ 𝑋)
1514adantl 273 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝑡) ∈ 𝑋)
16 xmetcl 12416 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋 ∧ (1st𝑡) ∈ 𝑋) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
1711, 13, 15, 16syl3anc 1199 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
183adantr 272 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
19 xp2nd 6030 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (2nd𝐶) ∈ 𝑌)
209, 19syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝐶) ∈ 𝑌)
21 xp2nd 6030 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (2nd𝑡) ∈ 𝑌)
2221adantl 273 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝑡) ∈ 𝑌)
23 xmetcl 12416 . . . . . . . 8 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌 ∧ (2nd𝑡) ∈ 𝑌) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
2418, 20, 22, 23syl3anc 1199 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
25 xrmaxcl 10961 . . . . . . 7 ((((1st𝐶)𝑀(1st𝑡)) ∈ ℝ* ∧ ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*) → sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*)
2617, 24, 25syl2anc 406 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*)
27 fveq2 5387 . . . . . . . . . 10 (𝑢 = 𝐶 → (1st𝑢) = (1st𝐶))
28 fveq2 5387 . . . . . . . . . 10 (𝑣 = 𝑡 → (1st𝑣) = (1st𝑡))
2927, 28oveqan12d 5759 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝐶)𝑀(1st𝑡)))
30 fveq2 5387 . . . . . . . . . 10 (𝑢 = 𝐶 → (2nd𝑢) = (2nd𝐶))
31 fveq2 5387 . . . . . . . . . 10 (𝑣 = 𝑡 → (2nd𝑣) = (2nd𝑡))
3230, 31oveqan12d 5759 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝐶)𝑁(2nd𝑡)))
3329, 32preq12d 3576 . . . . . . . 8 ((𝑢 = 𝐶𝑣 = 𝑡) → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))})
3433supeq1d 6840 . . . . . . 7 ((𝑢 = 𝐶𝑣 = 𝑡) → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3534, 1ovmpoga 5866 . . . . . 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 1199 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝐶𝑃𝑡) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3736breq1d 3907 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) < 𝑅))
386adantr 272 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑅 ∈ ℝ*)
39 xrmaxltsup 10967 . . . . 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 1199 . . . 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 2646 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅} = {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)})
43 1st2nd2 6039 . . . . . . 7 (𝑛 ∈ (𝑋 × 𝑌) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
4443ad2antrl 479 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
45 xp1st 6029 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (1st𝑛) ∈ 𝑋)
4645ad2antrl 479 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ 𝑋)
47 simprrl 511 . . . . . . 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 12455 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋𝑅 ∈ ℝ*) → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
502, 48, 6, 49syl3anc 1199 . . . . . . . 8 (𝜑 → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
5150adantr 272 . . . . . . 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 911 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
53 xp2nd 6030 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (2nd𝑛) ∈ 𝑌)
5453ad2antrl 479 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ 𝑌)
55 simprrr 512 . . . . . . 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 12455 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌𝑅 ∈ ℝ*) → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
583, 56, 6, 57syl3anc 1199 . . . . . . . 8 (𝜑 → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
5958adantr 272 . . . . . . 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 911 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))
6144, 52, 60jca32 306 . . . . 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 503 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
63 simprrl 511 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
6450adantr 272 . . . . . . . . . 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 512 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))
6858adantr 272 . . . . . . . . . 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 306 . . . . . . 7 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ 𝑋 ∧ (2nd𝑛) ∈ 𝑌)))
72 elxp6 6033 . . . . . . 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 306 . . . . 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 568 . . . 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 5387 . . . . . . . 8 (𝑡 = 𝑛 → (1st𝑡) = (1st𝑛))
7978oveq2d 5756 . . . . . . 7 (𝑡 = 𝑛 → ((1st𝐶)𝑀(1st𝑡)) = ((1st𝐶)𝑀(1st𝑛)))
8079breq1d 3907 . . . . . 6 (𝑡 = 𝑛 → (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ↔ ((1st𝐶)𝑀(1st𝑛)) < 𝑅))
81 fveq2 5387 . . . . . . . 8 (𝑡 = 𝑛 → (2nd𝑡) = (2nd𝑛))
8281oveq2d 5756 . . . . . . 7 (𝑡 = 𝑛 → ((2nd𝐶)𝑁(2nd𝑡)) = ((2nd𝐶)𝑁(2nd𝑛)))
8382breq1d 3907 . . . . . 6 (𝑡 = 𝑛 → (((2nd𝐶)𝑁(2nd𝑡)) < 𝑅 ↔ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))
8480, 83anbi12d 462 . . . . 5 (𝑡 = 𝑛 → ((((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅) ↔ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
8584elrab 2811 . . . 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 6033 . . . 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 2113 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅)))
898, 42, 883eqtrd 2152 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 1314  wcel 1463  {crab 2395  {cpr 3496  cop 3498   class class class wbr 3897   × cxp 4505  cfv 5091  (class class class)co 5740  cmpo 5742  1st c1st 6002  2nd c2nd 6003  supcsup 6835  *cxr 7763   < clt 7764  ∞Metcxmet 12044  ballcbl 12046
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-map 6510  df-sup 6837  df-inf 6838  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-xneg 9499  df-xadd 9500  df-seqfrec 10159  df-exp 10233  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-topgen 12036  df-psmet 12051  df-xmet 12052  df-bl 12054  df-mopn 12055  df-top 12060  df-topon 12073  df-bases 12105
This theorem is referenced by:  xmettxlem  12573  xmettx  12574
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