Theorem caubl 25280

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 6901	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑛 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑛)))
3	2	sseq1d 4008	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑛)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
4	3	imbi2d 339	. . . . . . . . . . 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 6901	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
6	5	sseq1d 4008	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
7	6	imbi2d 339	. . . . . . . . . . 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 6901	. . . . . . . . . . . . 13 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
9	8	sseq1d 4008	. . . . . . . . . . . 12 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))))
10	9	imbi2d 339	. . . . . . . . . . 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 3999	. . . . . . . . . . . 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 12935	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
15		fvoveq1 7442	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
16	15	fveq2d 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17		2fveq3 6901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
18	16, 17	sseq12d 4010	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
19	18	rspccva 3605	. . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝑛))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
21	20	anassrs 466	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
22		sstr2 3983	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 12923	. . . . . . . . . 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 5696	. . . . . . . . . . . . . . . 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	ffvelcdmd 7094	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) ∈ (𝑋 × ℝ+))
34		1st2nd 8044	. . . . . . . . . . . . . . . 16 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑛) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
35	29, 33, 34	sylancr 585	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑛) = ⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
36	35	fveq2d 6900	. . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . 14 ⊢ ((1st ‘(𝐹‘𝑛))(ball‘𝐷)(2nd ‘(𝐹‘𝑛))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛))⟩)
38	36, 37	eqtr4di 2783	. . . . . . . . . . . . 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 8026	. . . . . . . . . . . . . . 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 8027	. . . . . . . . . . . . . . . 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 13052	. . . . . . . . . . . . . 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 13052	. . . . . . . . . . . . . 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 13017	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ → (2nd ‘(𝐹‘𝑛)) ∈ ℝ)
50		rpre 13017	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ)
51		ltle 11334	. . . . . . . . . . . . . . . . 17 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
52	49, 50, 51	syl2an 594	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘(𝐹‘𝑛)) ∈ ℝ+ ∧ 𝑟 ∈ ℝ+) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → (2nd ‘(𝐹‘𝑛)) ≤ 𝑟))
53	44, 46, 52	syl2anc 582	. . . . . . . . . . . . . . 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 24373	. . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . 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 4015	. . . . . . . . . . . 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 3983	. . . . . . . . . . . 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 578	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → 𝑘 ∈ ℕ)
62	31, 61	ffvelcdmd 7094	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ+))
63		xp1st 8026	. . . . . . . . . . . . . 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 8027	. . . . . . . . . . . . . 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 24363	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 8044	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑋 × ℝ+) ∧ (𝐹‘𝑘) ∈ (𝑋 × ℝ+)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
70	29, 62, 69	sylancr 585	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
71	70	fveq2d 6900	. . . . . . . . . . . . 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 7422	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
73	71, 72	eqtr4di 2783	. . . . . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) ∧ 𝑘 ∈ (ℤ≥‘𝑛)) → (1st ‘(𝐹‘𝑘)) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
75		ssel 3970	. . . . . . . . . . 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 24340	. . . . . . . . . . 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 837	. . . . . . . . . 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 411	. . . . . . . 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 3134	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑛 ∈ ℕ ∧ (2nd ‘(𝐹‘𝑛)) < 𝑟)) → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟)
83	82	expr 455	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑛 ∈ ℕ) → ((2nd ‘(𝐹‘𝑛)) < 𝑟 → ∀𝑘 ∈ (ℤ≥‘𝑛)((1st ‘(𝐹‘𝑛))𝐷(1st ‘(𝐹‘𝑘))) < 𝑟))
84	83	reximdva 3157	. . . 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 3156	. . 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 12898	. . 3 ⊢ ℕ = (ℤ≥‘1)
88		1zzd 12626	. . 3 ⊢ (𝜑 → 1 ∈ ℤ)
89		fvco3 6996	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
90	30, 89	sylan 578	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
91		fvco3 6996	. . . 4 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
92	30, 91	sylan 578	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑛) = (1st ‘(𝐹‘𝑛)))
93		1stcof 8024	. . . 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 25252	. 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 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   ⊆ wss 3944  ⟨cop 4636   class class class wbr 5149   × cxp 5676   ∘ ccom 5682  Rel wrel 5683  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  1^st c1st 7992  2^nd c2nd 7993  ℝcr 11139  1c1 11141   + caddc 11143  ℝ^*cxr 11279   < clt 11280   ≤ cle 11281  ℕcn 12245  ℤcz 12591  ℤ_≥cuz 12855  ℝ⁺crp 13009  ∞Metcxmet 21281  ballcbl 21283  Cauccau 25225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-psmet 21288  df-xmet 21289  df-bl 21291  df-cau 25228

This theorem is referenced by:  bcthlem4  25299  heiborlem9  37420