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Theorem caubl 23152
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 fveq2 6229 . . . . . . . . . . . . . 14 (𝑟 = 𝑛 → (𝐹𝑟) = (𝐹𝑛))
32fveq2d 6233 . . . . . . . . . . . . 13 (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑛)))
43sseq1d 3665 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
54imbi2d 329 . . . . . . . . . . 11 (𝑟 = 𝑛 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
6 fveq2 6229 . . . . . . . . . . . . . 14 (𝑟 = 𝑘 → (𝐹𝑟) = (𝐹𝑘))
76fveq2d 6233 . . . . . . . . . . . . 13 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
87sseq1d 3665 . . . . . . . . . . . 12 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
98imbi2d 329 . . . . . . . . . . 11 (𝑟 = 𝑘 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
10 fveq2 6229 . . . . . . . . . . . . . 14 (𝑟 = (𝑘 + 1) → (𝐹𝑟) = (𝐹‘(𝑘 + 1)))
1110fveq2d 6233 . . . . . . . . . . . . 13 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
1211sseq1d 3665 . . . . . . . . . . . 12 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
1312imbi2d 329 . . . . . . . . . . 11 (𝑟 = (𝑘 + 1) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
14 ssid 3657 . . . . . . . . . . . 12 ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))
15142a1i 12 . . . . . . . . . . 11 (𝑛 ∈ ℤ → ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
16 caubl.4 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
17 eluznn 11796 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
18 oveq1 6697 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
1918fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
2019fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
21 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐹𝑛) = (𝐹𝑘))
2221fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
2320, 22sseq12d 3667 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
2423rspccva 3339 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2516, 17, 24syl2an 493 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2625anassrs 681 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
27 sstr2 3643 . . . . . . . . . . . . . 14 (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
2826, 27syl 17 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
2928expcom 450 . . . . . . . . . . . 12 (𝑘 ∈ (ℤ𝑛) → ((𝜑𝑛 ∈ ℕ) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
3029a2d 29 . . . . . . . . . . 11 (𝑘 ∈ (ℤ𝑛) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) → ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
315, 9, 13, 9, 15, 30uzind4 11784 . . . . . . . . . 10 (𝑘 ∈ (ℤ𝑛) → ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
3231com12 32 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝑘 ∈ (ℤ𝑛) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
3332ad2ant2r 798 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
34 relxp 5160 . . . . . . . . . . . . . . . 16 Rel (𝑋 × ℝ+)
35 caubl.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
3635ad3antrrr 766 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
37 simplrl 817 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑛 ∈ ℕ)
3836, 37ffvelrnd 6400 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) ∈ (𝑋 × ℝ+))
39 1st2nd 7258 . . . . . . . . . . . . . . . 16 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑛) ∈ (𝑋 × ℝ+)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
4034, 38, 39sylancr 696 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
4140fveq2d 6233 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
42 df-ov 6693 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
4341, 42syl6eqr 2703 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))))
44 caubl.2 . . . . . . . . . . . . . . 15 (𝜑𝐷 ∈ (∞Met‘𝑋))
4544ad3antrrr 766 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
46 xp1st 7242 . . . . . . . . . . . . . . 15 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑛)) ∈ 𝑋)
4738, 46syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑛)) ∈ 𝑋)
48 xp2nd 7243 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4938, 48syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
5049rpxrd 11911 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
51 simpllr 815 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ+)
5251rpxrd 11911 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ*)
53 simplrr 818 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) < 𝑟)
54 rpre 11877 . . . . . . . . . . . . . . . . 17 ((2nd ‘(𝐹𝑛)) ∈ ℝ+ → (2nd ‘(𝐹𝑛)) ∈ ℝ)
55 rpre 11877 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
56 ltle 10164 . . . . . . . . . . . . . . . . 17 (((2nd ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5754, 55, 56syl2an 493 . . . . . . . . . . . . . . . 16 (((2nd ‘(𝐹𝑛)) ∈ ℝ+𝑟 ∈ ℝ+) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5849, 51, 57syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5953, 58mpd 15 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ≤ 𝑟)
60 ssbl 22275 . . . . . . . . . . . . . 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))
6145, 47, 50, 52, 59, 60syl221anc 1377 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
6243, 61eqsstrd 3672 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
63 sstr2 3643 . . . . . . . . . . . 12 (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → (((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
6462, 63syl5com 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)))
65 simprl 809 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
6665, 17sylan 487 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
6736, 66ffvelrnd 6400 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
68 xp1st 7242 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
6967, 68syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
70 xp2nd 7243 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
7167, 70syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
72 blcntr 22265 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
7345, 69, 71, 72syl3anc 1366 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
74 1st2nd 7258 . . . . . . . . . . . . . . 15 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑘) ∈ (𝑋 × ℝ+)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7534, 67, 74sylancr 696 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7675fveq2d 6233 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
77 df-ov 6693 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7876, 77syl6eqr 2703 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
7973, 78eleqtrrd 2733 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
80 ssel 3630 . . . . . . . . . . 11 (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) → ((1st ‘(𝐹𝑘)) ∈ ((ball‘𝐷)‘(𝐹𝑘)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
8164, 79, 80syl6ci 71 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟)))
82 elbl2 22242 . . . . . . . . . . 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))
8345, 52, 47, 69, 82syl22anc 1367 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟) ↔ ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8481, 83sylibd 229 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8584ex 449 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)))
8633, 85mpdd 43 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8786ralrimiv 2994 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
8887expr 642 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8988reximdva 3046 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
9089ralimdva 2991 . . 3 (𝜑 → (∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
911, 90mpd 15 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
92 nnuz 11761 . . 3 ℕ = (ℤ‘1)
93 1zzd 11446 . . 3 (𝜑 → 1 ∈ ℤ)
94 fvco3 6314 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
9535, 94sylan 487 . . 3 ((𝜑𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
96 fvco3 6314 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
9735, 96sylan 487 . . 3 ((𝜑𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
98 1stcof 7240 . . . 4 (𝐹:ℕ⟶(𝑋 × ℝ+) → (1st𝐹):ℕ⟶𝑋)
9935, 98syl 17 . . 3 (𝜑 → (1st𝐹):ℕ⟶𝑋)
10092, 44, 93, 95, 97, 99iscauf 23124 . 2 (𝜑 → ((1st𝐹) ∈ (Cau‘𝐷) ↔ ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
10191, 100mpbird 247 1 (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  cop 4216   class class class wbr 4685   × cxp 5141  ccom 5147  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  cr 9973  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cn 11058  cz 11415  cuz 11725  +crp 11870  ∞Metcxmt 19779  ballcbl 19781  Caucca 23097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-psmet 19786  df-xmet 19787  df-bl 19789  df-cau 23100
This theorem is referenced by:  bcthlem4  23170  heiborlem9  33748
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