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Theorem caublcls 23107

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 ∧ (1^st ∘ 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 2622	. 2 ⊢ (ℤ_≥‘𝐴) = (ℤ_≥‘𝐴)
2		caubl.2	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
3	2	3ad2ant1 1082	. . 3 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4		caublcls.6	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
5	4	mopntopon 22244	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6	3, 5	syl 17	. 2 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7		simp3 1063	. . 3 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
8	7	nnzd 11481	. 2 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9		simp2 1062	. 2 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃)
10		fveq2 6191	. . . . . . . . 9 ⊢ (𝑟 = 𝐴 → (𝐹‘𝑟) = (𝐹‘𝐴))
11	10	fveq2d 6195	. . . . . . . 8 ⊢ (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝐴)))
12	11	sseq1d 3632	. . . . . . 7 ⊢ (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
13	12	imbi2d 330	. . . . . 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)))))
14		fveq2 6191	. . . . . . . . 9 ⊢ (𝑟 = 𝑘 → (𝐹‘𝑟) = (𝐹‘𝑘))
15	14	fveq2d 6195	. . . . . . . 8 ⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
16	15	sseq1d 3632	. . . . . . 7 ⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
17	16	imbi2d 330	. . . . . 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)))))
18		fveq2 6191	. . . . . . . . 9 ⊢ (𝑟 = (𝑘 + 1) → (𝐹‘𝑟) = (𝐹‘(𝑘 + 1)))
19	18	fveq2d 6195	. . . . . . . 8 ⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
20	19	sseq1d 3632	. . . . . . 7 ⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
21	20	imbi2d 330	. . . . . 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)))))
22		ssid 3624	. . . . . . 7 ⊢ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))
23	22	2a1i 12	. . . . . 6 ⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
24		caubl.4	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)))
25		eluznn 11758	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → 𝑘 ∈ ℕ)
26		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
27	26	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
28	27	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
29		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
30	29	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘)))
31	28, 30	sseq12d 3634	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))))
32	31	rspccva 3308	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
33	24, 25, 32	syl2an 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
34	33	anassrs 680	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))
35		sstr2 3610	. . . . . . . . 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))))
36	34, 35	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))))
37	36	expcom 451	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ_≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))))
38	37	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)))))
39	13, 17, 21, 17, 23, 38	uzind4 11746	. . . . 5 ⊢ (𝑘 ∈ (ℤ_≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))
40	39	impcom 446	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))
41	40	3adantl2 1218	. . 3 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))
42	3	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
43		simpl1 1064	. . . . . . . 8 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → 𝜑)
44		caubl.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 × ℝ⁺))
45	43, 44	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → 𝐹:ℕ⟶(𝑋 × ℝ⁺))
46	25	3ad2antl3 1225	. . . . . . 7 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → 𝑘 ∈ ℕ)
47	45, 46	ffvelrnd 6360	. . . . . 6 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑘) ∈ (𝑋 × ℝ⁺))
48		xp1st 7198	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝐹‘𝑘)) ∈ 𝑋)
49	47, 48	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → (1^st ‘(𝐹‘𝑘)) ∈ 𝑋)
50		xp2nd 7199	. . . . . 6 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ⁺)
51	47, 50	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → (2^nd ‘(𝐹‘𝑘)) ∈ ℝ⁺)
52		blcntr 22218	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝐹‘𝑘)) ∈ 𝑋 ∧ (2^nd ‘(𝐹‘𝑘)) ∈ ℝ⁺) → (1^st ‘(𝐹‘𝑘)) ∈ ((1^st ‘(𝐹‘𝑘))(ball‘𝐷)(2^nd ‘(𝐹‘𝑘))))
53	42, 49, 51, 52	syl3anc 1326	. . . 4 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → (1^st ‘(𝐹‘𝑘)) ∈ ((1^st ‘(𝐹‘𝑘))(ball‘𝐷)(2^nd ‘(𝐹‘𝑘))))
54		fvco3 6275	. . . . 5 ⊢ ((𝐹:ℕ⟶(𝑋 × ℝ⁺) ∧ 𝑘 ∈ ℕ) → ((1^st ∘ 𝐹)‘𝑘) = (1^st ‘(𝐹‘𝑘)))
55	45, 46, 54	syl2anc 693	. . . 4 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((1^st ∘ 𝐹)‘𝑘) = (1^st ‘(𝐹‘𝑘)))
56		1st2nd2 7205	. . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ⁺) → (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
57	47, 56	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑘) = ⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
58	57	fveq2d 6195	. . . . 5 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((ball‘𝐷)‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩))
59		df-ov 6653	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑘))(ball‘𝐷)(2^nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘⟨(1^st ‘(𝐹‘𝑘)), (2^nd ‘(𝐹‘𝑘))⟩)
60	58, 59	syl6eqr 2674	. . . 4 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((1^st ‘(𝐹‘𝑘))(ball‘𝐷)(2^nd ‘(𝐹‘𝑘))))
61	53, 55, 60	3eltr4d 2716	. . 3 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((1^st ∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝑘)))
62	41, 61	sseldd 3604	. 2 ⊢ (((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝐴)) → ((1^st ∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝐴)))
63	44	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 × ℝ⁺))
64	63	3adant2 1080	. . . . . 6 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 × ℝ⁺))
65		1st2nd2 7205	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ⁺) → (𝐹‘𝐴) = ⟨(1^st ‘(𝐹‘𝐴)), (2^nd ‘(𝐹‘𝐴))⟩)
66	64, 65	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) = ⟨(1^st ‘(𝐹‘𝐴)), (2^nd ‘(𝐹‘𝐴))⟩)
67	66	fveq2d 6195	. . . 4 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((ball‘𝐷)‘⟨(1^st ‘(𝐹‘𝐴)), (2^nd ‘(𝐹‘𝐴))⟩))
68		df-ov 6653	. . . 4 ⊢ ((1^st ‘(𝐹‘𝐴))(ball‘𝐷)(2^nd ‘(𝐹‘𝐴))) = ((ball‘𝐷)‘⟨(1^st ‘(𝐹‘𝐴)), (2^nd ‘(𝐹‘𝐴))⟩)
69	67, 68	syl6eqr 2674	. . 3 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((1^st ‘(𝐹‘𝐴))(ball‘𝐷)(2^nd ‘(𝐹‘𝐴))))
70		xp1st 7198	. . . . 5 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝐹‘𝐴)) ∈ 𝑋)
71	64, 70	syl 17	. . . 4 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1^st ‘(𝐹‘𝐴)) ∈ 𝑋)
72		xp2nd 7199	. . . . . 6 ⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝐹‘𝐴)) ∈ ℝ⁺)
73	64, 72	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2^nd ‘(𝐹‘𝐴)) ∈ ℝ⁺)
74	73	rpxrd 11873	. . . 4 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2^nd ‘(𝐹‘𝐴)) ∈ ℝ^*)
75		blssm 22223	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝐹‘𝐴)) ∈ 𝑋 ∧ (2^nd ‘(𝐹‘𝐴)) ∈ ℝ^*) → ((1^st ‘(𝐹‘𝐴))(ball‘𝐷)(2^nd ‘(𝐹‘𝐴))) ⊆ 𝑋)
76	3, 71, 74, 75	syl3anc 1326	. . 3 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((1^st ‘(𝐹‘𝐴))(ball‘𝐷)(2^nd ‘(𝐹‘𝐴))) ⊆ 𝑋)
77	69, 76	eqsstrd 3639	. 2 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ 𝑋)
78	1, 6, 8, 9, 62, 77	lmcls 21106	1 ⊢ ((𝜑 ∧ (1^st ∘ 𝐹)(⇝_𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ⟨cop 4183 class class class wbr 4653 × cxp 5112 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 1c1 9937 + caddc 9939 ℝ^*cxr 10073 ℕcn 11020 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 ∞Metcxmt 19731 ballcbl 19733 MetOpencmopn 19736 TopOnctopon 20715 clsccl 20822 ⇝_𝑡clm 21030

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-lm 21033

This theorem is referenced by: bcthlem3 23123 heiborlem8 33617

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