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Theorem caubl 25280
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 6901 . . . . . . . . . . . . 13 (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑛)))
32sseq1d 4008 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
43imbi2d 339 . . . . . . . . . . 11 (𝑟 = 𝑛 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
5 2fveq3 6901 . . . . . . . . . . . . 13 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
65sseq1d 4008 . . . . . . . . . . . 12 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
76imbi2d 339 . . . . . . . . . . 11 (𝑟 = 𝑘 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
8 2fveq3 6901 . . . . . . . . . . . . 13 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
98sseq1d 4008 . . . . . . . . . . . 12 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
109imbi2d 339 . . . . . . . . . . 11 (𝑟 = (𝑘 + 1) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
11 ssid 3999 . . . . . . . . . . . 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 12935 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
15 fvoveq1 7442 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
1615fveq2d 6900 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17 2fveq3 6901 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
1816, 17sseq12d 4010 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
1918rspccva 3605 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2013, 14, 19syl2an 594 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2120anassrs 466 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
22 sstr2 3983 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 12923 . . . . . . . . . 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 745 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
29 relxp 5696 . . . . . . . . . . . . . . . 16 Rel (𝑋 × ℝ+)
30 caubl.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
3130ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32 simplrl 775 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑛 ∈ ℕ)
3331, 32ffvelcdmd 7094 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) ∈ (𝑋 × ℝ+))
34 1st2nd 8044 . . . . . . . . . . . . . . . 16 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑛) ∈ (𝑋 × ℝ+)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3529, 33, 34sylancr 585 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3635fveq2d 6900 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
37 df-ov 7422 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3836, 37eqtr4di 2783 . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41 xp1st 8026 . . . . . . . . . . . . . . 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 8027 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4433, 43syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4544rpxrd 13052 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
46 simpllr 774 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ+)
4746rpxrd 13052 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ*)
48 simplrr 776 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) < 𝑟)
49 rpre 13017 . . . . . . . . . . . . . . . . 17 ((2nd ‘(𝐹𝑛)) ∈ ℝ+ → (2nd ‘(𝐹𝑛)) ∈ ℝ)
50 rpre 13017 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
51 ltle 11334 . . . . . . . . . . . . . . . . 17 (((2nd ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5249, 50, 51syl2an 594 . . . . . . . . . . . . . . . 16 (((2nd ‘(𝐹𝑛)) ∈ ℝ+𝑟 ∈ ℝ+) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5344, 46, 52syl2anc 582 . . . . . . . . . . . . . . 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 24373 . . . . . . . . . . . . . 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 1378 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
5738, 56eqsstrd 4015 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
58 sstr2 3983 . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
6160, 14sylan 578 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
6231, 61ffvelcdmd 7094 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
63 xp1st 8026 . . . . . . . . . . . . . 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 8027 . . . . . . . . . . . . . 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 24363 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
6840, 64, 66, 67syl3anc 1368 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
69 1st2nd 8044 . . . . . . . . . . . . . . 15 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑘) ∈ (𝑋 × ℝ+)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7029, 62, 69sylancr 585 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7170fveq2d 6900 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
72 df-ov 7422 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7371, 72eqtr4di 2783 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
7468, 73eleqtrrd 2828 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
75 ssel 3970 . . . . . . . . . . 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 24340 . . . . . . . . . . 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 837 . . . . . . . . . 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 238 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8079ex 411 . . . . . . . 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 3134 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
8382expr 455 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8483reximdva 3157 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8584ralimdva 3156 . . 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 12898 . . 3 ℕ = (ℤ‘1)
88 1zzd 12626 . . 3 (𝜑 → 1 ∈ ℤ)
89 fvco3 6996 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
9030, 89sylan 578 . . 3 ((𝜑𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
91 fvco3 6996 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
9230, 91sylan 578 . . 3 ((𝜑𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
93 1stcof 8024 . . . 4 (𝐹:ℕ⟶(𝑋 × ℝ+) → (1st𝐹):ℕ⟶𝑋)
9430, 93syl 17 . . 3 (𝜑 → (1st𝐹):ℕ⟶𝑋)
9587, 39, 88, 90, 92, 94iscauf 25252 . 2 (𝜑 → ((1st𝐹) ∈ (Cau‘𝐷) ↔ ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
9686, 95mpbird 256 1 (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3944  cop 4636   class class class wbr 5149   × cxp 5676  ccom 5682  Rel wrel 5683  wf 6545  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  cr 11139  1c1 11141   + caddc 11143  *cxr 11279   < clt 11280  cle 11281  cn 12245  cz 12591  cuz 12855  +crp 13009  ∞Metcxmet 21281  ballcbl 21283  Cauccau 25225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-psmet 21288  df-xmet 21289  df-bl 21291  df-cau 25228
This theorem is referenced by:  bcthlem4  25299  heiborlem9  37420
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