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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({((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 13674 . . 3 (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
5 xmetxpbl.c . . 3 (𝜑𝐶 ∈ (𝑋 × 𝑌))
6 xmetxpbl.r . . 3 (𝜑𝑅 ∈ ℝ*)
7 blval 13556 . . 3 ((𝑃 ∈ (∞Met‘(𝑋 × 𝑌)) ∧ 𝐶 ∈ (𝑋 × 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
84, 5, 6, 7syl3anc 1238 . 2 (𝜑 → (𝐶(ball‘𝑃)𝑅) = {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅})
95adantr 276 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝐶 ∈ (𝑋 × 𝑌))
10 simpr 110 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑡 ∈ (𝑋 × 𝑌))
112adantr 276 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
12 xp1st 6160 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (1st𝐶) ∈ 𝑋)
139, 12syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝐶) ∈ 𝑋)
14 xp1st 6160 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (1st𝑡) ∈ 𝑋)
1514adantl 277 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (1st𝑡) ∈ 𝑋)
16 xmetcl 13519 . . . . . . . 8 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋 ∧ (1st𝑡) ∈ 𝑋) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
1711, 13, 15, 16syl3anc 1238 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((1st𝐶)𝑀(1st𝑡)) ∈ ℝ*)
183adantr 276 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
19 xp2nd 6161 . . . . . . . . 9 (𝐶 ∈ (𝑋 × 𝑌) → (2nd𝐶) ∈ 𝑌)
209, 19syl 14 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝐶) ∈ 𝑌)
21 xp2nd 6161 . . . . . . . . 9 (𝑡 ∈ (𝑋 × 𝑌) → (2nd𝑡) ∈ 𝑌)
2221adantl 277 . . . . . . . 8 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (2nd𝑡) ∈ 𝑌)
23 xmetcl 13519 . . . . . . . 8 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌 ∧ (2nd𝑡) ∈ 𝑌) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
2418, 20, 22, 23syl3anc 1238 . . . . . . 7 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*)
25 xrmaxcl 11244 . . . . . . 7 ((((1st𝐶)𝑀(1st𝑡)) ∈ ℝ* ∧ ((2nd𝐶)𝑁(2nd𝑡)) ∈ ℝ*) → sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*)
2617, 24, 25syl2anc 411 . . . . . 6 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) ∈ ℝ*)
27 fveq2 5511 . . . . . . . . . 10 (𝑢 = 𝐶 → (1st𝑢) = (1st𝐶))
28 fveq2 5511 . . . . . . . . . 10 (𝑣 = 𝑡 → (1st𝑣) = (1st𝑡))
2927, 28oveqan12d 5888 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((1st𝑢)𝑀(1st𝑣)) = ((1st𝐶)𝑀(1st𝑡)))
30 fveq2 5511 . . . . . . . . . 10 (𝑢 = 𝐶 → (2nd𝑢) = (2nd𝐶))
31 fveq2 5511 . . . . . . . . . 10 (𝑣 = 𝑡 → (2nd𝑣) = (2nd𝑡))
3230, 31oveqan12d 5888 . . . . . . . . 9 ((𝑢 = 𝐶𝑣 = 𝑡) → ((2nd𝑢)𝑁(2nd𝑣)) = ((2nd𝐶)𝑁(2nd𝑡)))
3329, 32preq12d 3676 . . . . . . . 8 ((𝑢 = 𝐶𝑣 = 𝑡) → {((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))} = {((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))})
3433supeq1d 6980 . . . . . . 7 ((𝑢 = 𝐶𝑣 = 𝑡) → sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3534, 1ovmpoga 5998 . . . . . 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 1238 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (𝐶𝑃𝑡) = sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ))
3736breq1d 4010 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) < 𝑅))
386adantr 276 . . . . 5 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → 𝑅 ∈ ℝ*)
39 xrmaxltsup 11250 . . . . 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 1238 . . . 4 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → (sup({((1st𝐶)𝑀(1st𝑡)), ((2nd𝐶)𝑁(2nd𝑡))}, ℝ*, < ) < 𝑅 ↔ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)))
4137, 40bitrd 188 . . 3 ((𝜑𝑡 ∈ (𝑋 × 𝑌)) → ((𝐶𝑃𝑡) < 𝑅 ↔ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)))
4241rabbidva 2725 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (𝐶𝑃𝑡) < 𝑅} = {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)})
43 1st2nd2 6170 . . . . . . 7 (𝑛 ∈ (𝑋 × 𝑌) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
4443ad2antrl 490 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
45 xp1st 6160 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (1st𝑛) ∈ 𝑋)
4645ad2antrl 490 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ 𝑋)
47 simprrl 539 . . . . . . 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 13558 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ (1st𝐶) ∈ 𝑋𝑅 ∈ ℝ*) → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
502, 48, 6, 49syl3anc 1238 . . . . . . . 8 (𝜑 → ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ↔ ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅)))
5150adantr 276 . . . . . . 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 944 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
53 xp2nd 6161 . . . . . . . 8 (𝑛 ∈ (𝑋 × 𝑌) → (2nd𝑛) ∈ 𝑌)
5453ad2antrl 490 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ 𝑌)
55 simprrr 540 . . . . . . 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 13558 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ (2nd𝐶) ∈ 𝑌𝑅 ∈ ℝ*) → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
583, 56, 6, 57syl3anc 1238 . . . . . . . 8 (𝜑 → ((2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅) ↔ ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
5958adantr 276 . . . . . . 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 944 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (𝑋 × 𝑌) ∧ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))) → (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))
6144, 52, 60jca32 310 . . . . 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 529 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → 𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩)
63 simprrl 539 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅))
6450adantr 276 . . . . . . . . . 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 147 . . . . . . . . 9 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((1st𝑛) ∈ 𝑋 ∧ ((1st𝐶)𝑀(1st𝑛)) < 𝑅))
6665simpld 112 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (1st𝑛) ∈ 𝑋)
67 simprrr 540 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅))
6858adantr 276 . . . . . . . . . 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 147 . . . . . . . . 9 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((2nd𝑛) ∈ 𝑌 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))
7069simpld 112 . . . . . . . 8 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (2nd𝑛) ∈ 𝑌)
7162, 66, 70jca32 310 . . . . . . 7 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ 𝑋 ∧ (2nd𝑛) ∈ 𝑌)))
72 elxp6 6164 . . . . . . 7 (𝑛 ∈ (𝑋 × 𝑌) ↔ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ 𝑋 ∧ (2nd𝑛) ∈ 𝑌)))
7371, 72sylibr 134 . . . . . 6 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → 𝑛 ∈ (𝑋 × 𝑌))
7465simprd 114 . . . . . 6 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((1st𝐶)𝑀(1st𝑛)) < 𝑅)
7569simprd 114 . . . . . 6 ((𝜑 ∧ (𝑛 = ⟨(1st𝑛), (2nd𝑛)⟩ ∧ ((1st𝑛) ∈ ((1st𝐶)(ball‘𝑀)𝑅) ∧ (2nd𝑛) ∈ ((2nd𝐶)(ball‘𝑁)𝑅)))) → ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)
7673, 74, 75jca32 310 . . . . 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 596 . . . 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 5511 . . . . . . . 8 (𝑡 = 𝑛 → (1st𝑡) = (1st𝑛))
7978oveq2d 5885 . . . . . . 7 (𝑡 = 𝑛 → ((1st𝐶)𝑀(1st𝑡)) = ((1st𝐶)𝑀(1st𝑛)))
8079breq1d 4010 . . . . . 6 (𝑡 = 𝑛 → (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ↔ ((1st𝐶)𝑀(1st𝑛)) < 𝑅))
81 fveq2 5511 . . . . . . . 8 (𝑡 = 𝑛 → (2nd𝑡) = (2nd𝑛))
8281oveq2d 5885 . . . . . . 7 (𝑡 = 𝑛 → ((2nd𝐶)𝑁(2nd𝑡)) = ((2nd𝐶)𝑁(2nd𝑛)))
8382breq1d 4010 . . . . . 6 (𝑡 = 𝑛 → (((2nd𝐶)𝑁(2nd𝑡)) < 𝑅 ↔ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅))
8480, 83anbi12d 473 . . . . 5 (𝑡 = 𝑛 → ((((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅) ↔ (((1st𝐶)𝑀(1st𝑛)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑛)) < 𝑅)))
8584elrab 2893 . . . 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 6164 . . . 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 223 . . 3 (𝜑 → (𝑛 ∈ {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} ↔ 𝑛 ∈ (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅))))
8887eqrdv 2175 . 2 (𝜑 → {𝑡 ∈ (𝑋 × 𝑌) ∣ (((1st𝐶)𝑀(1st𝑡)) < 𝑅 ∧ ((2nd𝐶)𝑁(2nd𝑡)) < 𝑅)} = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅)))
898, 42, 883eqtrd 2214 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 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  1st c1st 6133  2nd c2nd 6134  supcsup 6975  *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
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