MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caubl Structured version   Visualization version   GIF version

Theorem caubl 25342
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 6911 . . . . . . . . . . . . 13 (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑛)))
32sseq1d 4015 . . . . . . . . . . . 12 (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
43imbi2d 340 . . . . . . . . . . 11 (𝑟 = 𝑛 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
5 2fveq3 6911 . . . . . . . . . . . . 13 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
65sseq1d 4015 . . . . . . . . . . . 12 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
76imbi2d 340 . . . . . . . . . . 11 (𝑟 = 𝑘 → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
8 2fveq3 6911 . . . . . . . . . . . . 13 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
98sseq1d 4015 . . . . . . . . . . . 12 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
109imbi2d 340 . . . . . . . . . . 11 (𝑟 = (𝑘 + 1) → (((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))) ↔ ((𝜑𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))))
11 ssid 4006 . . . . . . . . . . . 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 12960 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑘 ∈ ℕ)
15 fvoveq1 7454 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
1615fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17 2fveq3 6911 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
1816, 17sseq12d 4017 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
1918rspccva 3621 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
22 sstr2 3990 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 12948 . . . . . . . . . 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 747 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ𝑛) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛))))
29 relxp 5703 . . . . . . . . . . . . . . . 16 Rel (𝑋 × ℝ+)
30 caubl.3 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
3130ad3antrrr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32 simplrl 777 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑛 ∈ ℕ)
3331, 32ffvelcdmd 7105 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑛) ∈ (𝑋 × ℝ+))
34 1st2nd 8064 . . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩))
37 df-ov 7434 . . . . . . . . . . . . . 14 ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑛)), (2nd ‘(𝐹𝑛))⟩)
3836, 37eqtr4di 2795 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41 xp1st 8046 . . . . . . . . . . . . . . 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 8047 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4433, 43syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ+)
4544rpxrd 13078 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) ∈ ℝ*)
46 simpllr 776 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ+)
4746rpxrd 13078 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → 𝑟 ∈ ℝ*)
48 simplrr 778 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (2nd ‘(𝐹𝑛)) < 𝑟)
49 rpre 13043 . . . . . . . . . . . . . . . . 17 ((2nd ‘(𝐹𝑛)) ∈ ℝ+ → (2nd ‘(𝐹𝑛)) ∈ ℝ)
50 rpre 13043 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ ℝ+𝑟 ∈ ℝ)
51 ltle 11349 . . . . . . . . . . . . . . . . 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 24433 . . . . . . . . . . . . . 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 1383 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((1st ‘(𝐹𝑛))(ball‘𝐷)(2nd ‘(𝐹𝑛))) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
5738, 56eqsstrd 4018 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑛)) ⊆ ((1st ‘(𝐹𝑛))(ball‘𝐷)𝑟))
58 sstr2 3990 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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, 61ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
63 xp1st 8046 . . . . . . . . . . . . . 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 8047 . . . . . . . . . . . . . 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 24423 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
6840, 64, 66, 67syl3anc 1373 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
69 1st2nd 8064 . . . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . 13 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
72 df-ov 7434 . . . . . . . . . . . . 13 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
7371, 72eqtr4di 2795 . . . . . . . . . . . 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 3977 . . . . . . . . . . 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 24400 . . . . . . . . . . 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 839 . . . . . . . . . 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 239 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ𝑛)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) → ((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8079ex 412 . . . . . . . 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 3145 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟)
8382expr 456 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8483reximdva 3168 . . . 4 ((𝜑𝑟 ∈ ℝ+) → (∃𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
8584ralimdva 3167 . . 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 12921 . . 3 ℕ = (ℤ‘1)
88 1zzd 12648 . . 3 (𝜑 → 1 ∈ ℤ)
89 fvco3 7008 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
9030, 89sylan 580 . . 3 ((𝜑𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
91 fvco3 7008 . . . 4 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
9230, 91sylan 580 . . 3 ((𝜑𝑛 ∈ ℕ) → ((1st𝐹)‘𝑛) = (1st ‘(𝐹𝑛)))
93 1stcof 8044 . . . 4 (𝐹:ℕ⟶(𝑋 × ℝ+) → (1st𝐹):ℕ⟶𝑋)
9430, 93syl 17 . . 3 (𝜑 → (1st𝐹):ℕ⟶𝑋)
9587, 39, 88, 90, 92, 94iscauf 25314 . 2 (𝜑 → ((1st𝐹) ∈ (Cau‘𝐷) ↔ ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)((1st ‘(𝐹𝑛))𝐷(1st ‘(𝐹𝑘))) < 𝑟))
9686, 95mpbird 257 1 (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  cop 4632   class class class wbr 5143   × cxp 5683  ccom 5689  Rel wrel 5690  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  cr 11154  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cn 12266  cz 12613  cuz 12878  +crp 13034  ∞Metcxmet 21349  ballcbl 21351  Cauccau 25287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-psmet 21356  df-xmet 21357  df-bl 21359  df-cau 25290
This theorem is referenced by:  bcthlem4  25361  heiborlem9  37826
  Copyright terms: Public domain W3C validator