Theorem caublcls 25610

Description: The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-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)))
caublcls.6	⊢ 𝐽 = (MetOpen‘𝐷)

Assertion

Ref	Expression
caublcls	⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴))))

Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑛,𝑋

Allowed substitution hints:   𝜑(𝑛)   𝐴(𝑛)   𝑃(𝑛)   𝐽(𝑛)

Proof of Theorem caublcls

Dummy variables 𝑘 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. 2 ⊢ (ℤ≥‘𝐴) = (ℤ≥‘𝐴)
2		caubl.2	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
3	2	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4		caublcls.6	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
5	4	mopntopon 24738	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	syl 18	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7		simp3 1156	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
8	7	nnzd 12700	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9		simp2 1155	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃)
10		2fveq3 6882	. . . . . . . 8 ⊢ (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝐴)))
11	10	sseq1d 3962	. . . . . . 7 ⊢ (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
12	11	imbi2d 343	. . . . . 6 ⊢ (𝑟 = 𝐴 → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))))
13		2fveq3 6882	. . . . . . . 8 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
14	13	sseq1d 3962	. . . . . . 7 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
15	14	imbi2d 343	. . . . . 6 ⊢ (𝑟 = 𝑘 → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))))
16		2fveq3 6882	. . . . . . . 8 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17	16	sseq1d 3962	. . . . . . 7 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
18	17	imbi2d 343	. . . . . 6 ⊢ (𝑟 = (𝑘 + 1) → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))))
19		ssid 3953	. . . . . . 7 ⊢ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))
20	19	2a1i 12	. . . . . 6 ⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
21		caubl.4	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))
22		eluznn 13026	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝑘 ∈ ℕ)
23		fvoveq1 7435	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
24	23	fveq2d 6881	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
25		2fveq3 6882	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
26	24, 25	sseq12d 3964	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
27	26	rspccva 3576	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
28	21, 22, 27	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
29	28	anassrs 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
30		sstr2 3938	. . . . . . . . 9 ⊢ (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
31	29, 30	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
32	31	expcom 419	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))))
33	32	a2d 30	. . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝐴) → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))))
34	12, 15, 18, 15, 20, 33	uzind4 13014	. . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
35	34	impcom 413	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))
36	35	3adantl2 1186	. . 3 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))
37	3	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
38		simpl1 1210	. . . . . . . 8 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝜑)
39		caubl.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ+))
40	38, 39	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
41	22	3ad2antl3 1206	. . . . . . 7 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝑘 ∈ ℕ)
42	40, 41	ffvelcdmd 7077	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ+))
43		xp1st 8022	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹‘𝑘)) ∈ 𝑋)
44	42, 43	syl 18	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st ‘(𝐹‘𝑘)) ∈ 𝑋)
45		xp2nd 8023	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹‘𝑘)) ∈ ℝ+)
46	42, 45	syl 18	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (2nd ‘(𝐹‘𝑘)) ∈ ℝ+)
47		blcntr 24712	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹‘𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝑘)) ∈ ℝ+) → (1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))))
48	37, 44, 46, 47	syl3anc 1398	. . . 4 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))))
49		fvco3 6977	. . . . 5 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
50	40, 41, 49	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
51		1st2nd2 8029	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
52	42, 51	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
53	52	fveq2d 6881	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩))
54		df-ov 7415	. . . . 5 ⊢ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
55	53, 54	eqtr4di 2814	. . . 4 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))))
56	48, 50, 55	3eltr4d 2876	. . 3 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st ∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
57	36, 56	sseldd 3932	. 2 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st ∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝐴)))
58	39	ffvelcdmda 7076	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 × ℝ+))
59	58	3adant2 1149	. . . . . 6 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 × ℝ+))
60		1st2nd2 8029	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (𝐹‘𝐴) = ⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩)
61	59, 60	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) = ⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩)
62	61	fveq2d 6881	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩))
63		df-ov 7415	. . . 4 ⊢ ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩)
64	62, 63	eqtr4di 2814	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))))
65		xp1st 8022	. . . . 5 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹‘𝐴)) ∈ 𝑋)
66	59, 65	syl 18	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st ‘(𝐹‘𝐴)) ∈ 𝑋)
67		xp2nd 8023	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹‘𝐴)) ∈ ℝ+)
68	59, 67	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd ‘(𝐹‘𝐴)) ∈ ℝ+)
69	68	rpxrd 13146	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd ‘(𝐹‘𝐴)) ∈ ℝ*)
70		blssm 24717	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹‘𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝐴)) ∈ ℝ*) → ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋)
71	3, 66, 69, 70	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋)
72	64, 71	eqsstrd 3965	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ 𝑋)
73	1, 6, 8, 9, 57, 72	lmcls 23600	1 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  ⟨cop 4590   class class class wbr 5103   × cxp 5649   ∘ ccom 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  1^st c1st 7988  2^nd c2nd 7989  1c1 11182   + caddc 11184  ℝ^*cxr 11323  ℕcn 12316  ℤcz 12674  ℤ_≥cuz 12946  ℝ⁺crp 13101  ∞Metcxmet 21643  ballcbl 21645  MetOpencmopn 21648  TopOnctopon 23208  clsccl 23316  ⇝_𝑡clm 23524

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-xneg 13222  df-xadd 13223  df-xmul 13224  df-topgen 17594  df-psmet 21650  df-xmet 21651  df-bl 21653  df-mopn 21654  df-top 23192  df-topon 23209  df-bases 23244  df-cld 23317  df-ntr 23318  df-cls 23319  df-lm 23527

This theorem is used by:  bcthlem3  25627  heiborlem8  38720