Theorem caubl 23913

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 6677	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑛)))
3	2	sseq1d 4000	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
4	3	imbi2d 343	. . . . . . . . . . 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 6677	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
6	5	sseq1d 4000	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
7	6	imbi2d 343	. . . . . . . . . . 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 6677	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
9	8	sseq1d 4000	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
10	9	imbi2d 343	. . . . . . . . . . 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 3991	. . . . . . . . . . . 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 12321	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
15		fvoveq1 7181	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
16	15	fveq2d 6676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17		2fveq3 6677	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
18	16, 17	sseq12d 4002	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
19	18	rspccva 3624	. . . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
21	20	anassrs 470	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
22		sstr2 3976	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 12309	. . . . . . . . . 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 745	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ≥‘𝑛) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
29		relxp 5575	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑋 × ℝ+)
30		caubl.3	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))
31	30	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32		simplrl 775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ ℕ)
33	31, 32	ffvelrnd 6854	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) ∈ (𝑋 × ℝ+))
34		1st2nd 7740	. . . . . . . . . . . . . . . 16 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑛) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
35	29, 33, 34	sylancr 589	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
36	35	fveq2d 6676	. . . . . . . . . . . . . 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 7161	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
38	36, 37	syl6eqr 2876	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41		xp1st 7723	. . . . . . . . . . . . . . 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 7724	. . . . . . . . . . . . . . . 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 12435	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
46		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℝ+)
47	46	rpxrd 12435	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℝ*)
48		simplrr 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) < 𝑟)
49		rpre 12400	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
50		rpre 12400	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
51		ltle 10731	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
52	49, 50, 51	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ ∧ 𝑟 ∈ ℝ+) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
53	44, 46, 52	syl2anc 586	. . . . . . . . . . . . . . 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 23035	. . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . 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 4007	. . . . . . . . . . . 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 3976	. . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
61	60, 14	sylan 582	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
62	31, 61	ffvelrnd 6854	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ+))
63		xp1st 7723	. . . . . . . . . . . . . 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 7724	. . . . . . . . . . . . . 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 23025	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 7740	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑘) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
70	29, 62, 69	sylancr 589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
71	70	fveq2d 6676	. . . . . . . . . . . . 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 7161	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
73	71, 72	syl6eqr 2876	. . . . . . . . . . . 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 2918	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
75		ssel 3963	. . . . . . . . . . 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 23002	. . . . . . . . . . 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 241	. . . . . . . . 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 415	. . . . . . . 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 3183	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟)
83	82	expr 459	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
84	83	reximdva 3276	. . . 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 3179	. . 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 12284	. . 3 ⊢ ℕ = (ℤ≥‘1)
88		1zzd 12016	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		fvco3 6762	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
90	30, 89	sylan 582	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
91		fvco3 6762	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
92	30, 91	sylan 582	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
93		1stcof 7721	. . . 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 23885	. 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 259	1 ⊢ (𝜑 → (1st ∘ 𝐹) ∈ (Cau‘𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938  ⟨cop 4575   class class class wbr 5068   × cxp 5555   ∘ ccom 5561  Rel wrel 5562  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  ℝcr 10538  1c1 10540   + caddc 10542  ℝ^*cxr 10676   < clt 10677   ≤ cle 10678  ℕcn 11640  ℤcz 11984  ℤ_≥cuz 12246  ℝ⁺crp 12392  ∞Metcxmet 20532  ballcbl 20534  Cauccau 23858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-psmet 20539  df-xmet 20540  df-bl 20542  df-cau 23861

This theorem is referenced by:  bcthlem4  23932  heiborlem9  35099