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Theorem caubl 24483
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 6776 . . . . . . . . . . . . 13 (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑛)))
32sseq1d 3957 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
43imbi2d 341 . . . . . . . . . . 11 (𝑟 = 𝑛 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
5 2fveq3 6776 . . . . . . . . . . . . 13 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
65sseq1d 3957 . . . . . . . . . . . 12 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
76imbi2d 341 . . . . . . . . . . 11 (𝑟 = 𝑘 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
8 2fveq3 6776 . . . . . . . . . . . . 13 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
98sseq1d 3957 . . . . . . . . . . . 12 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
109imbi2d 341 . . . . . . . . . . 11 (𝑟 = (𝑘 + 1) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
11 ssid 3948 . . . . . . . . . . . 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 12669 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
15 fvoveq1 7295 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
1615fveq2d 6775 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17 2fveq3 6776 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
1816, 17sseq12d 3959 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
1918rspccva 3560 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2013, 14, 19syl2an 596 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2120anassrs 468 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
22 sstr2 3933 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 12657 . . . . . . . . . 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 5608 . . . . . . . . . . . . . . . 16 Rel (𝑋 × ℝ+)
30 caubl.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
3130ad3antrrr 727 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32 simplrl 774 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑛 ∈ ℕ)
3331, 32ffvelrnd 6959 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) ∈ (𝑋 × ℝ+))
34 1st2nd 7874 . . . . . . . . . . . . . . . 16 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑛) ∈ (𝑋 × ℝ+)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3529, 33, 34sylancr 587 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) = ⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3635fveq2d 6775 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
37 df-ov 7275 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3836, 37eqtr4di 2798 . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41 xp1st 7857 . . . . . . . . . . . . . . 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 7858 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4433, 43syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4544rpxrd 12784 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
46 simpllr 773 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ+)
4746rpxrd 12784 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ*)
48 simplrr 775 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) < 𝑟)
49 rpre 12749 . . . . . . . . . . . . . . . . 17 ((2nd ‘(𝐹𝑛)) ∈ ℝ+ → (2nd ‘(𝐹𝑛)) ∈ ℝ)
50 rpre 12749 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
51 ltle 11074 . . . . . . . . . . . . . . . . 17 (((2nd ‘(𝐹𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5249, 50, 51syl2an 596 . . . . . . . . . . . . . . . 16 (((2nd ‘(𝐹𝑛)) ∈ ℝ+𝑟 ∈ ℝ+) → ((2nd ‘(𝐹𝑛)) < 𝑟 → (2nd ‘(𝐹𝑛)) ≤ 𝑟))
5344, 46, 52syl2anc 584 . . . . . . . . . . . . . . 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 23587 . . . . . . . . . . . . . 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 1380 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
5738, 56eqsstrd 3964 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
58 sstr2 3933 . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
6160, 14sylan 580 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
6231, 61ffvelrnd 6959 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
63 xp1st 7857 . . . . . . . . . . . . . 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 7858 . . . . . . . . . . . . . 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 23577 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
6840, 64, 66, 67syl3anc 1370 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
69 1st2nd 7874 . . . . . . . . . . . . . . 15 ((Rel (𝑋 × ℝ+) ∧ (𝐹𝑘) ∈ (𝑋 × ℝ+)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7029, 62, 69sylancr 587 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7170fveq2d 6775 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
72 df-ov 7275 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7371, 72eqtr4di 2798 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
7468, 73eleqtrrd 2844 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
75 ssel 3919 . . . . . . . . . . 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 23554 . . . . . . . . . . 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 836 . . . . . . . . . 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 413 . . . . . . . 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 3109 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
8382expr 457 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8483reximdva 3205 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8584ralimdva 3105 . . 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 12632 . . 3 ℕ = (ℤ‘1)
88 1zzd 12362 . . 3 (𝜑 → 1 ∈ ℤ)
89 fvco3 6864 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
9030, 89sylan 580 . . 3 ((𝜑𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
91 fvco3 6864 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
9230, 91sylan 580 . . 3 ((𝜑𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
93 1stcof 7855 . . . 4 (𝐹:ℕ⟶(𝑋 × ℝ+) → (1st𝐹):ℕ⟶𝑋)
9430, 93syl 17 . . 3 (𝜑 → (1st𝐹):ℕ⟶𝑋)
9587, 39, 88, 90, 92, 94iscauf 24455 . 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 396   = wceq 1542  wcel 2110  wral 3066  wrex 3067  wss 3892  cop 4573   class class class wbr 5079   × cxp 5588  ccom 5594  Rel wrel 5595  wf 6428  cfv 6432  (class class class)co 7272  1st c1st 7823  2nd c2nd 7824  cr 10881  1c1 10883   + caddc 10885  *cxr 11019   < clt 11020  cle 11021  cn 11984  cz 12330  cuz 12593  +crp 12741  ∞Metcxmet 20593  ballcbl 20595  Cauccau 24428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-cnex 10938  ax-resscn 10939  ax-1cn 10940  ax-icn 10941  ax-addcl 10942  ax-addrcl 10943  ax-mulcl 10944  ax-mulrcl 10945  ax-mulcom 10946  ax-addass 10947  ax-mulass 10948  ax-distr 10949  ax-i2m1 10950  ax-1ne0 10951  ax-1rid 10952  ax-rnegex 10953  ax-rrecex 10954  ax-cnre 10955  ax-pre-lttri 10956  ax-pre-lttrn 10957  ax-pre-ltadd 10958  ax-pre-mulgt0 10959
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7229  df-ov 7275  df-oprab 7276  df-mpo 7277  df-om 7708  df-1st 7825  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-er 8490  df-map 8609  df-pm 8610  df-en 8726  df-dom 8727  df-sdom 8728  df-pnf 11022  df-mnf 11023  df-xr 11024  df-ltxr 11025  df-le 11026  df-sub 11218  df-neg 11219  df-div 11644  df-nn 11985  df-2 12047  df-n0 12245  df-z 12331  df-uz 12594  df-rp 12742  df-xneg 12859  df-xadd 12860  df-xmul 12861  df-psmet 20600  df-xmet 20601  df-bl 20603  df-cau 24431
This theorem is referenced by:  bcthlem4  24502  heiborlem9  35986
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