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Theorem caubl 23318
Description: Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
caubl.2 (𝜑𝐷 ∈ (∞Met‘𝑋))
caubl.3 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
caubl.4 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
caubl.5 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟)
Assertion
Ref Expression
caubl (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
Distinct variable groups:   𝑛,𝑟,𝐷   𝑛,𝐹,𝑟   𝜑,𝑟   𝑛,𝑋,𝑟   𝜑,𝑛

Proof of Theorem caubl
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 caubl.5 . . 3 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟)
2 2fveq3 6413 . . . . . . . . . . . . 13 (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑛)))
32sseq1d 3829 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
43imbi2d 331 . . . . . . . . . . 11 (𝑟 = 𝑛 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
5 2fveq3 6413 . . . . . . . . . . . . 13 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
65sseq1d 3829 . . . . . . . . . . . 12 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
76imbi2d 331 . . . . . . . . . . 11 (𝑟 = 𝑘 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
8 2fveq3 6413 . . . . . . . . . . . . 13 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
98sseq1d 3829 . . . . . . . . . . . 12 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
109imbi2d 331 . . . . . . . . . . 11 (𝑟 = (𝑘 + 1) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
11 ssid 3820 . . . . . . . . . . . 12 ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))
12112a1i 12 . . . . . . . . . . 11 (𝑛 ∈ ℤ → ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
13 caubl.4 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
14 eluznn 11977 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
15 fvoveq1 6897 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
1615fveq2d 6412 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17 2fveq3 6413 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
1816, 17sseq12d 3831 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
1918rspccva 3501 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2013, 14, 19syl2an 585 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2120anassrs 455 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
22 sstr2 3805 . . . . . . . . . . . . . 14 (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
2321, 22syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
2423expcom 400 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑛) → ((𝜑𝑛 ∈ ℕ) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
2524a2d 29 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑛) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) → ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
264, 7, 10, 7, 12, 25uzind4 11964 . . . . . . . . . 10 (𝑘 ∈ (ℤ𝑛) → ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
2726com12 32 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘 ∈ (ℤ𝑛) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
2827ad2ant2r 744 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
29 relxp 5328 . . . . . . . . . . . . . . . 16 Rel (𝑋 × ℝ+)
30 caubl.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
3130ad3antrrr 712 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32 simplrl 786 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑛 ∈ ℕ)
3331, 32ffvelrnd 6582 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) ∈ (𝑋 × ℝ+))
34 1st2nd 7446 . . . . . . . . . . . . . . . 16 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑛) ∈ (𝑋 × ℝ+)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3529, 33, 34sylancr 577 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3635fveq2d 6412 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
37 df-ov 6877 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3836, 37syl6eqr 2858 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))))
39 caubl.2 . . . . . . . . . . . . . . 15 (𝜑𝐷 ∈ (∞Met‘𝑋))
4039ad3antrrr 712 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41 xp1st 7430 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑛)) ∈ 𝑋)
4233, 41syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑛)) ∈ 𝑋)
43 xp2nd 7431 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4433, 43syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4544rpxrd 12087 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
46 simpllr 784 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ+)
4746rpxrd 12087 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ*)
48 simplrr 787 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) < 𝑟)
49 rpre 12053 . . . . . . . . . . . . . . . . 17 ((2nd ‘(𝐹𝑛)) ∈ ℝ+ → (2nd ‘(𝐹𝑛)) ∈ ℝ)
50 rpre 12053 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
51 ltle 10411 . . . . . . . . . . . . . . . . 17 (((2nd ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5249, 50, 51syl2an 585 . . . . . . . . . . . . . . . 16 (((2nd ‘(𝐹𝑛)) ∈ ℝ+𝑟 ∈ ℝ+) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5344, 46, 52syl2anc 575 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5448, 53mpd 15 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ≤ 𝑟)
55 ssbl 22441 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑛)) ∈ 𝑋) ∧ ((2nd ‘(𝐹𝑛)) ∈ ℝ*𝑟 ∈ ℝ*) ∧ (2nd ‘(𝐹𝑛)) ≤ 𝑟) → ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
5640, 42, 45, 47, 54, 55syl221anc 1493 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
5738, 56eqsstrd 3836 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
58 sstr2 3805 . . . . . . . . . . . 12 (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → (((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
5957, 58syl5com 31 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
60 simprl 778 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
6160, 14sylan 571 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
6231, 61ffvelrnd 6582 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
63 xp1st 7430 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
6462, 63syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
65 xp2nd 7431 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
6662, 65syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
67 blcntr 22431 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
6840, 64, 66, 67syl3anc 1483 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
69 1st2nd 7446 . . . . . . . . . . . . . . 15 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑘) ∈ (𝑋 × ℝ+)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7029, 62, 69sylancr 577 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7170fveq2d 6412 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
72 df-ov 6877 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7371, 72syl6eqr 2858 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
7468, 73eleqtrrd 2888 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
75 ssel 3792 . . . . . . . . . . 11 (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) → ((1st ‘(𝐹𝑘)) ∈ ((ball‘𝐷)‘(𝐹𝑘)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
7659, 74, 75syl6ci 71 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
77 elbl2 22408 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑟 ∈ ℝ*) ∧ ((1st ‘(𝐹𝑛)) ∈ 𝑋 ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋)) → ((1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) ↔ ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
7840, 47, 42, 64, 77syl22anc 858 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) ↔ ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
7976, 78sylibd 230 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8079ex 399 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)))
8128, 80mpdd 43 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8281ralrimiv 3153 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
8382expr 446 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8483reximdva 3204 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8584ralimdva 3150 . . 3 (𝜑 → (∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
861, 85mpd 15 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
87 nnuz 11941 . . 3 ℕ = (ℤ‘1)
88 1zzd 11674 . . 3 (𝜑 → 1 ∈ ℤ)
89 fvco3 6496 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
9030, 89sylan 571 . . 3 ((𝜑𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
91 fvco3 6496 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
9230, 91sylan 571 . . 3 ((𝜑𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
93 1stcof 7428 . . . 4 (𝐹:ℕ⟶(𝑋 × ℝ+) → (1st𝐹):ℕ⟶𝑋)
9430, 93syl 17 . . 3 (𝜑 → (1st𝐹):ℕ⟶𝑋)
9587, 39, 88, 90, 92, 94iscauf 23290 . 2 (𝜑 → ((1st𝐹) ∈ (Cau‘𝐷) ↔ ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
9686, 95mpbird 248 1 (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wrex 3097  wss 3769  cop 4376   class class class wbr 4844   × cxp 5309  ccom 5315  Rel wrel 5316  wf 6097  cfv 6101  (class class class)co 6874  1st c1st 7396  2nd c2nd 7397  cr 10220  1c1 10222   + caddc 10224  *cxr 10358   < clt 10359  cle 10360  cn 11305  cz 11643  cuz 11904  +crp 12046  ∞Metcxmt 19939  ballcbl 19941  Caucca 23263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-n0 11560  df-z 11644  df-uz 11905  df-rp 12047  df-xneg 12162  df-xadd 12163  df-xmul 12164  df-psmet 19946  df-xmet 19947  df-bl 19949  df-cau 23266
This theorem is referenced by:  bcthlem4  23336  heiborlem9  33929
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