Theorem caublcls 25261

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 2735	. 2 ⊢ (ℤ≥‘𝐴) = (ℤ≥‘𝐴)
2		caubl.2	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4		caublcls.6	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
5	4	mopntopon 24378	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	syl 17	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7		simp3 1138	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
8	7	nnzd 12615	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9		simp2 1137	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃)
10		2fveq3 6881	. . . . . . . 8 ⊢ (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝐴)))
11	10	sseq1d 3990	. . . . . . 7 ⊢ (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
12	11	imbi2d 340	. . . . . 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 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
14	13	sseq1d 3990	. . . . . . 7 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
15	14	imbi2d 340	. . . . . 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 6881	. . . . . . . 8 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
17	16	sseq1d 3990	. . . . . . 7 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
18	17	imbi2d 340	. . . . . 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 3981	. . . . . . 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 12934	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝑘 ∈ ℕ)
23		fvoveq1 7428	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
24	23	fveq2d 6880	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
25		2fveq3 6881	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
26	24, 25	sseq12d 3992	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
27	26	rspccva 3600	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
29	28	anassrs 467	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
30		sstr2 3965	. . . . . . . . 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 17	. . . . . . . 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 413	. . . . . . 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 29	. . . . . 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 12922	. . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
35	34	impcom 407	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))
36	35	3adantl2 1168	. . 3 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))
37	3	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
38		simpl1 1192	. . . . . . . 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 17	. . . . . . 7 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
41	22	3ad2antl3 1188	. . . . . . 7 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝑘 ∈ ℕ)
42	40, 41	ffvelcdmd 7075	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ+))
43		xp1st 8020	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹‘𝑘)) ∈ 𝑋)
44	42, 43	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st ‘(𝐹‘𝑘)) ∈ 𝑋)
45		xp2nd 8021	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹‘𝑘)) ∈ ℝ+)
46	42, 45	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (2nd ‘(𝐹‘𝑘)) ∈ ℝ+)
47		blcntr 24352	. . . . 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 1373	. . . 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 6978	. . . . 5 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
50	40, 41, 49	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘)))
51		1st2nd2 8027	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
52	42, 51	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) = ⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
53	52	fveq2d 6880	. . . . 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 7408	. . . . 5 ⊢ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))⟩)
55	53, 54	eqtr4di 2788	. . . 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 2849	. . 3 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st ∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
57	36, 56	sseldd 3959	. 2 ⊢ (((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st ∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝐴)))
58	39	ffvelcdmda 7074	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 × ℝ+))
59	58	3adant2 1131	. . . . . 6 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 × ℝ+))
60		1st2nd2 8027	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (𝐹‘𝐴) = ⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩)
61	59, 60	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) = ⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩)
62	61	fveq2d 6880	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩))
63		df-ov 7408	. . . 4 ⊢ ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))⟩)
64	62, 63	eqtr4di 2788	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))))
65		xp1st 8020	. . . . 5 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹‘𝐴)) ∈ 𝑋)
66	59, 65	syl 17	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st ‘(𝐹‘𝐴)) ∈ 𝑋)
67		xp2nd 8021	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹‘𝐴)) ∈ ℝ+)
68	59, 67	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd ‘(𝐹‘𝐴)) ∈ ℝ+)
69	68	rpxrd 13052	. . . 4 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd ‘(𝐹‘𝐴)) ∈ ℝ*)
70		blssm 24357	. . . 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 1373	. . 3 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋)
72	64, 71	eqsstrd 3993	. 2 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ 𝑋)
73	1, 6, 8, 9, 57, 72	lmcls 23240	1 ⊢ ((𝜑 ∧ (1st ∘ 𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ⟨cop 4607   class class class wbr 5119   × cxp 5652   ∘ ccom 5658  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  1^st c1st 7986  2^nd c2nd 7987  1c1 11130   + caddc 11132  ℝ^*cxr 11268  ℕcn 12240  ℤcz 12588  ℤ_≥cuz 12852  ℝ⁺crp 13008  ∞Metcxmet 21300  ballcbl 21302  MetOpencmopn 21305  TopOnctopon 22848  clsccl 22956  ⇝_𝑡clm 23164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-pm 8843  df-en 8960  df-dom 8961  df-sdom 8962  df-sup 9454  df-inf 9455  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-n0 12502  df-z 12589  df-uz 12853  df-q 12965  df-rp 13009  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-topgen 17457  df-psmet 21307  df-xmet 21308  df-bl 21310  df-mopn 21311  df-top 22832  df-topon 22849  df-bases 22884  df-cld 22957  df-ntr 22958  df-cls 22959  df-lm 23167

This theorem is referenced by:  bcthlem3  25278  heiborlem8  37842