Theorem caubl 24472

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	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
caubl.3	⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))
caubl.4	⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))
caubl.5	⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝐹‘𝑛)) < 𝑟)

Assertion

Ref	Expression
caubl	⊢ (𝜑 → (1st ∘ 𝐹) ∈ (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.

Step	Hyp	Ref	Expression
1		caubl.5	. . 3 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝐹‘𝑛)) < 𝑟)
2		2fveq3 6779	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑛)))
3	2	sseq1d 3952	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
4	3	imbi2d 341	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑛 → (((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))))
5		2fveq3 6779	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
6	5	sseq1d 3952	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
7	6	imbi2d 341	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))))
8		2fveq3 6779	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
9	8	sseq1d 3952	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
10	9	imbi2d 341	. . . . . . . . . . 11 ⊢ (𝑟 = (𝑘 + 1) → (((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) ↔ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))))
11		ssid 3943	. . . . . . . . . . . 12 ⊢ ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))
12	11	2a1i 12	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
13		caubl.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))
14		eluznn 12658	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
15		fvoveq1 7298	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
16	15	fveq2d 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17		2fveq3 6779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
18	16, 17	sseq12d 3954	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
19	18	rspccva 3560	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
20	13, 14, 19	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
21	20	anassrs 468	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
22		sstr2 3928	. . . . . . . . . . . . . 14 ⊢ (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
23	21, 22	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
24	23	expcom 414	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘𝑛) → ((𝜑 ∧ 𝑛 ∈ ℕ) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))))
25	24	a2d 29	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑛) → (((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) → ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))))
26	4, 7, 10, 7, 12, 25	uzind4 12646	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ≥‘𝑛) → ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
27	26	com12 32	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ (ℤ≥‘𝑛) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
28	27	ad2ant2r 744	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ≥‘𝑛) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
29		relxp 5607	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑋 × ℝ+)
30		caubl.3	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))
31	30	ad3antrrr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32		simplrl 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ ℕ)
33	31, 32	ffvelrnd 6962	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) ∈ (𝑋 × ℝ+))
34		1st2nd 7880	. . . . . . . . . . . . . . . 16 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑛) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
35	29, 33, 34	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
36	35	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩))
37		df-ov 7278	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
38	36, 37	eqtr4di 2796	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))))
39		caubl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
40	39	ad3antrrr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41		xp1st 7863	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹‘𝑛)) ∈ 𝑋)
42	33, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑛)) ∈ 𝑋)
43		xp2nd 7864	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ+)
44	33, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ+)
45	44	rpxrd 12773	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
46		simpllr 773	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℝ+)
47	46	rpxrd 12773	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℝ*)
48		simplrr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) < 𝑟)
49		rpre 12738	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
50		rpre 12738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
51		ltle 11063	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
52	49, 50, 51	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ ∧ 𝑟 ∈ ℝ+) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
53	44, 46, 52	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
54	48, 53	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟)
55		ssbl 23576	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹‘𝑛)) ∈ 𝑋) ∧ ((2nd ‘(𝐹‘𝑛)) ∈ ℝ* ∧ 𝑟 ∈ ℝ*) ∧ (2nd ‘(𝐹‘𝑛)) ≤ 𝑟) → ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟))
56	40, 42, 45, 47, 54, 55	syl221anc 1380	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟))
57	38, 56	eqsstrd 3959	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟))
58		sstr2 3928	. . . . . . . . . . . 12 ⊢ (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → (((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟)))
59	57, 58	syl5com 31	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟)))
60		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
61	60, 14	sylan 580	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
62	31, 61	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ+))
63		xp1st 7863	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹‘𝑘)) ∈ 𝑋)
64	62, 63	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ 𝑋)
65		xp2nd 7864	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹‘𝑘)) ∈ ℝ+)
66	62, 65	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑘)) ∈ ℝ+)
67		blcntr 23566	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹‘𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝑘)) ∈ ℝ+) → (1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))))
68	40, 64, 66, 67	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))))
69		1st2nd 7880	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑘) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
70	29, 62, 69	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
71	70	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩))
72		df-ov 7278	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
73	71, 72	eqtr4di 2796	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))))
74	68, 73	eleqtrrd 2842	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
75		ssel 3914	. . . . . . . . . . 11 ⊢ (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟) → ((1st ‘(𝐹‘𝑘)) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)) → (1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟)))
76	59, 74, 75	syl6ci 71	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟)))
77		elbl2 23543	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑟 ∈ ℝ*) ∧ ((1st ‘(𝐹‘𝑛)) ∈ 𝑋 ∧ (1st ‘(𝐹‘𝑘)) ∈ 𝑋)) → ((1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟) ↔ ((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
78	40, 47, 42, 64, 77	syl22anc 836	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)𝑟) ↔ ((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
79	76, 78	sylibd 238	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → ((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
80	79	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ≥‘𝑛) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) → ((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟)))
81	28, 80	mpdd 43	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ≥‘𝑛) → ((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
82	81	ralrimiv 3102	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟)
83	82	expr 457	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
84	83	reximdva 3203	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → (∃𝑛 ∈ ℕ (2nd ‘(𝐹‘𝑛)) < 𝑟 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
85	84	ralimdva 3108	. . 3 ⊢ (𝜑 → (∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ (2nd ‘(𝐹‘𝑛)) < 𝑟 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
86	1, 85	mpd 15	. 2 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟)
87		nnuz 12621	. . 3 ⊢ ℕ = (ℤ≥‘1)
88		1zzd 12351	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		fvco3 6867	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
90	30, 89	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
91		fvco3 6867	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
92	30, 91	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
93		1stcof 7861	. . . 4 ⊢ (𝐹:ℕ⟶(𝑋 × ℝ+) → (1st ∘ 𝐹):ℕ⟶𝑋)
94	30, 93	syl 17	. . 3 ⊢ (𝜑 → (1st ∘ 𝐹):ℕ⟶𝑋)
95	87, 39, 88, 90, 92, 94	iscauf 24444	. 2 ⊢ (𝜑 → ((1st ∘ 𝐹) ∈ (Cau‘𝐷) ↔ ∀𝑟 ∈ ℝ+ ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
96	86, 95	mpbird 256	1 ⊢ (𝜑 → (1st ∘ 𝐹) ∈ (Cau‘𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   × cxp 5587   ∘ ccom 5593  Rel wrel 5594  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  ℝcr 10870  1c1 10872   + caddc 10874  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010  ℕcn 11973  ℤcz 12319  ℤ_≥cuz 12582  ℝ⁺crp 12730  ∞Metcxmet 20582  ballcbl 20584  Cauccau 24417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-psmet 20589  df-xmet 20590  df-bl 20592  df-cau 24420

This theorem is referenced by:  bcthlem4  24491  heiborlem9  35977