Theorem caubl 25361

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 6925	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑛)))
3	2	sseq1d 4040	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
4	3	imbi2d 340	. . . . . . . . . . 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 6925	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
6	5	sseq1d 4040	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
7	6	imbi2d 340	. . . . . . . . . . 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 6925	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
9	8	sseq1d 4040	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
10	9	imbi2d 340	. . . . . . . . . . 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 4031	. . . . . . . . . . . 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 12983	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
15		fvoveq1 7471	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
16	15	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17		2fveq3 6925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
18	16, 17	sseq12d 4042	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
19	18	rspccva 3634	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
21	20	anassrs 467	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
22		sstr2 4015	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 12971	. . . . . . . . . 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 746	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → (𝑘 ∈ (ℤ≥‘𝑛) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
29		relxp 5718	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑋 × ℝ+)
30		caubl.3	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))
31	30	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
32		simplrl 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑛 ∈ ℕ)
33	31, 32	ffvelcdmd 7119	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) ∈ (𝑋 × ℝ+))
34		1st2nd 8080	. . . . . . . . . . . . . . . 16 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑛) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
35	29, 33, 34	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
36	35	fveq2d 6924	. . . . . . . . . . . . . 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 7451	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
38	36, 37	eqtr4di 2798	. . . . . . . . . . . . 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 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝐷 ∈ (∞Met‘𝑋))
41		xp1st 8062	. . . . . . . . . . . . . . 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 8063	. . . . . . . . . . . . . . . 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 13100	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) ∈ ℝ*)
46		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℝ+)
47	46	rpxrd 13100	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑟 ∈ ℝ*)
48		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (2nd ‘(𝐹‘𝑛)) < 𝑟)
49		rpre 13065	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
50		rpre 13065	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
51		ltle 11378	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
52	49, 50, 51	syl2an 595	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ ∧ 𝑟 ∈ ℝ+) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
53	44, 46, 52	syl2anc 583	. . . . . . . . . . . . . . 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 24454	. . . . . . . . . . . . . 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 1381	. . . . . . . . . . . . 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 4047	. . . . . . . . . . . 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 4015	. . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → 𝑛 ∈ ℕ)
61	60, 14	sylan 579	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
62	31, 61	ffvelcdmd 7119	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ+))
63		xp1st 8062	. . . . . . . . . . . . . 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 8063	. . . . . . . . . . . . . 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 24444	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 8080	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑘) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
70	29, 62, 69	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
71	70	fveq2d 6924	. . . . . . . . . . . . 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 7451	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
73	71, 72	eqtr4di 2798	. . . . . . . . . . . 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 2847	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
75		ssel 4002	. . . . . . . . . . 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 24421	. . . . . . . . . . 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 838	. . . . . . . . . 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 239	. . . . . . . . 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 412	. . . . . . . 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 3151	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟)
83	82	expr 456	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
84	83	reximdva 3174	. . . 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 3173	. . 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 12946	. . 3 ⊢ ℕ = (ℤ≥‘1)
88		1zzd 12674	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		fvco3 7021	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
90	30, 89	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
91		fvco3 7021	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
92	30, 91	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
93		1stcof 8060	. . . 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 25333	. 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 257	1 ⊢ (𝜑 → (1st ∘ 𝐹) ∈ (Cau‘𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ⟨cop 4654   class class class wbr 5166   × cxp 5698   ∘ ccom 5704  Rel wrel 5705  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  ℝcr 11183  1c1 11185   + caddc 11187  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  ℕcn 12293  ℤcz 12639  ℤ_≥cuz 12903  ℝ⁺crp 13057  ∞Metcxmet 21372  ballcbl 21374  Cauccau 25306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-psmet 21379  df-xmet 21380  df-bl 21382  df-cau 25309

This theorem is referenced by:  bcthlem4  25380  heiborlem9  37779