Theorem abelth 24195

Description: Abel's theorem. If the power series Σ𝑛 ∈ ℕ₀𝐴(𝑛)(𝑥↑𝑛) is convergent at 1, then it is equal to the limit from "below", along a Stolz angle 𝑆 (note that the 𝑀 = 1 case of a Stolz angle is the real line [0, 1]). (Continuity on 𝑆 ∖ {1} follows more generally from psercn 24180.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Hypotheses

Ref	Expression
abelth.1	⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
abelth.2	⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
abelth.3	⊢ (𝜑 → 𝑀 ∈ ℝ)
abelth.4	⊢ (𝜑 → 0 ≤ 𝑀)
abelth.5	⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
abelth.6	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))

Assertion

Ref	Expression
abelth	⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Distinct variable groups:   𝑥,𝑛,𝑧,𝑀   𝐴,𝑛,𝑥,𝑧   𝜑,𝑛,𝑥   𝑆,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑧)   𝑆(𝑧)   𝐹(𝑥,𝑧,𝑛)

Proof of Theorem abelth

Dummy variables 𝑗 𝑤 𝑦 𝑟 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abelth.1	. . . 4 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ)
2		abelth.2	. . . 4 ⊢ (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
3		abelth.3	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4		abelth.4	. . . 4 ⊢ (𝜑 → 0 ≤ 𝑀)
5		abelth.5	. . . 4 ⊢ 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
6		abelth.6	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
7	1, 2, 3, 4, 5, 6	abelthlem4 24188	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
8	1, 2, 3, 4, 5, 6	abelthlem9 24194	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟))
9	1, 2, 3, 4, 5	abelthlem2 24186	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
10	9	simpld 475	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ 𝑆)
11	10	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 1 ∈ 𝑆)
12		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
13	11, 12	ovresd 6801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14		ax-1cn 9994	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
15		ssrab2 3687	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} ⊆ ℂ
16	5, 15	eqsstri 3635	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 ⊆ ℂ
17	16, 12	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
18		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
19	18	cnmetdval 22574	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
20	14, 17, 19	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
21	13, 20	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
22	21	breq1d 4663	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
23	7	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
24	23, 11	ffvelrnd 6360	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) ∈ ℂ)
25	7	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
26	25	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
27	18	cnmetdval 22574	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘1) ∈ ℂ ∧ (𝐹‘𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
28	24, 26, 27	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
29	28	breq1d 4663	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟))
30	22, 29	imbi12d 334	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟) ↔ ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟)))
31	30	ralbidva 2985	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → (∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟) ↔ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟)))
32	31	rexbidv 3052	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → (∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟) ↔ ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟)))
33	8, 32	mpbird 247	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
34	33	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
35		cnxmet 22576	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
36		xmetres2 22166	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
37	35, 16, 36	mp2an 708	. . . . . . . . . 10 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
38	37	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
39	35	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
40		eqid 2622	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
41		eqid 2622	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
42	41	cnfldtopn 22585	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
43		eqid 2622	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
44	40, 42, 43	metrest 22329	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
45	35, 16, 44	mp2an 708	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
46	45, 42	metcnp 22346	. . . . . . . . 9 ⊢ ((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ 𝑆) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))))
47	38, 39, 10, 46	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))))
48	7, 34, 47	mpbir2and 957	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
49	48	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
50		simpr 477	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
51	50	fveq2d 6195	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
52	49, 51	eleqtrrd 2704	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
53		eldifsn 4317	. . . . . . 7 ⊢ (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1))
54	9	simprd 479	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
55		abscl 14018	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
56	55	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ)
57	56	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) ∈ ℝ))
58		absge0 14027	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → 0 ≤ (abs‘𝑤))
59	58	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 0 ≤ (abs‘𝑤))
60	59	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → 0 ≤ (abs‘𝑤)))
61	1, 2	abelthlem1 24185	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
62	61	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
63	56	rexrd 10089	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ*)
64		1re 10039	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
65		rexr 10085	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*)
66	64, 65	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ∈ ℝ*)
67		iccssxr 12256	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0[,]+∞) ⊆ ℝ*
68		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))
69		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) = sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )
70	68, 1, 69	radcnvcl 24171	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
71	67, 70	sseldi 3601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
72	71	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
73		xrltletr 11988	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((abs‘𝑤) ∈ ℝ* ∧ 1 ∈ ℝ* ∧ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) → (((abs‘𝑤) < 1 ∧ 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
74	63, 66, 72, 73	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (((abs‘𝑤) < 1 ∧ 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
75	62, 74	mpan2d 710	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
76	57, 60, 75	3jcad 1243	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → ((abs‘𝑤) ∈ ℝ ∧ 0 ≤ (abs‘𝑤) ∧ (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
77		0cn 10032	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
78	18	cnmetdval 22574	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
79	77, 78	mpan 706	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
80		abssub 14066	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
81	77, 80	mpan 706	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
82		subid1 10301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
83	82	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
84	79, 81, 83	3eqtrd 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
85	84	breq1d 4663	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
86	85	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
87		0re 10040	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
88		elico2 12237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ ∧ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) → ((abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) ↔ ((abs‘𝑤) ∈ ℝ ∧ 0 ≤ (abs‘𝑤) ∧ (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
89	87, 72, 88	sylancr 695	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) ↔ ((abs‘𝑤) ∈ ℝ ∧ 0 ≤ (abs‘𝑤) ∧ (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
90	76, 86, 89	3imtr4d 283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 → (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
91	90	imdistanda 729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1) → (𝑤 ∈ ℂ ∧ (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
92	64	rexri 10097	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ*
93		elbl 22193	. . . . . . . . . . . . . . . . . . 19 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1)))
94	35, 77, 92, 93	mp3an 1424	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
95		absf 14077	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
96		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
97		elpreima 6337	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs Fn ℂ → (𝑤 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↔ (𝑤 ∈ ℂ ∧ (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
98	95, 96, 97	mp2b 10	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↔ (𝑤 ∈ ℂ ∧ (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
99	91, 94, 98	3imtr4g 285	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) → 𝑤 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
100	99	ssrdv 3609	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0(ball‘(abs ∘ − ))1) ⊆ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
101	54, 100	sstrd 3613	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
102	101	resmptd 5452	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))))
103	6	reseq1i 5392	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1}))
104		difss 3737	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∖ {1}) ⊆ 𝑆
105		resmpt 5449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∖ {1}) ⊆ 𝑆 → ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))))
106	104, 105	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
107	103, 106	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
108	102, 107	syl6eqr 2674	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) = (𝐹 ↾ (𝑆 ∖ {1})))
109		cnvimass 5485	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
110	95	fdmi 6052	. . . . . . . . . . . . . . . . . . 19 ⊢ dom abs = ℂ
111	109, 110	sseqtri 3637	. . . . . . . . . . . . . . . . . 18 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
112	111	sseli 3599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → 𝑥 ∈ ℂ)
113	68	pserval2 24165	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗) = ((𝐴‘𝑗) · (𝑥↑𝑗)))
114	113	sumeq2dv 14433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗) = Σ𝑗 ∈ ℕ0 ((𝐴‘𝑗) · (𝑥↑𝑗)))
115		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗))
116		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑥↑𝑛) = (𝑥↑𝑗))
117	115, 116	oveq12d 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑗) · (𝑥↑𝑗)))
118	117	cbvsumv 14426	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ0 ((𝐴‘𝑗) · (𝑥↑𝑗))
119	114, 118	syl6reqr 2675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗))
120	112, 119	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗))
121	120	mpteq2ia 4740	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) = (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗))
122		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) = (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
123		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ if(sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑣) + sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑣) + 1)) = if(sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑣) + sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑣) + 1))
124	68, 121, 1, 69, 122, 123	psercn 24180	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ∈ ((◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ))
125		rescncf 22700	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∖ {1}) ⊆ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → ((𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ∈ ((◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ) → ((𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ)))
126	101, 124, 125	sylc 65	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
127	108, 126	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
128	127	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
129	104, 16	sstri 3612	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ {1}) ⊆ ℂ
130		ssid 3624	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
131		eqid 2622	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
132	41	cnfldtop 22587	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂfld) ∈ Top
133	41	cnfldtopon 22586	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
134	133	toponunii 20721	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
135	134	restid 16094	. . . . . . . . . . . . . . 15 ⊢ ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
136	132, 135	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
137	136	eqcomi 2631	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
138	41, 131, 137	cncfcn 22712	. . . . . . . . . . . 12 ⊢ (((𝑆 ∖ {1}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
139	129, 130, 138	mp2an 708	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld))
140	128, 139	syl6eleq 2711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
141		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
142		resttopon 20965	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑆 ∖ {1}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1})))
143	133, 129, 142	mp2an 708	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1}))
144	143	toponunii 20721	. . . . . . . . . . 11 ⊢ (𝑆 ∖ {1}) = ∪ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
145	144	cncnpi 21082	. . . . . . . . . 10 ⊢ (((𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)) ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
146	140, 141, 145	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
147		cnex 10017	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
148	147, 16	ssexi 4803	. . . . . . . . . . . 12 ⊢ 𝑆 ∈ V
149		restabs 20969	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆 ∧ 𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})))
150	132, 104, 148, 149	mp3an 1424	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
151	150	oveq1i 6660	. . . . . . . . . 10 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))
152	151	fveq1i 6192	. . . . . . . . 9 ⊢ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)
153	146, 152	syl6eleqr 2712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
154		resttop 20964	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
155	132, 148, 154	mp2an 708	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
156	155	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
157	104	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝑆 ∖ {1}) ⊆ 𝑆)
158	10	snssd 4340	. . . . . . . . . . . . 13 ⊢ (𝜑 → {1} ⊆ 𝑆)
159	41	cnfldhaus 22588	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂfld) ∈ Haus
160	134	sncld 21175	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂfld)))
161	159, 14, 160	mp2an 708	. . . . . . . . . . . . . 14 ⊢ {1} ∈ (Clsd‘(TopOpen‘ℂfld))
162	134	restcldi 20977	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
163	16, 161, 162	mp3an12 1414	. . . . . . . . . . . . 13 ⊢ ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
164	134	restuni 20966	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
165	132, 16, 164	mp2an 708	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
166	165	cldopn 20835	. . . . . . . . . . . . 13 ⊢ ({1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
167	158, 163, 166	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
168	165	isopn3 20870	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) → ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1})))
169	155, 104, 168	mp2an 708	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
170	167, 169	sylib 208	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
171	170	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
172	171	biimpar 502	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})))
173	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
174	165, 134	cnprest 21093	. . . . . . . . 9 ⊢ (((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) ∧ (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ∧ 𝐹:𝑆⟶ℂ)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) ↔ (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)))
175	156, 157, 172, 173, 174	syl22anc 1327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) ↔ (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)))
176	153, 175	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
177	53, 176	sylan2br 493	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
178	177	anassrs 680	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
179	52, 178	pm2.61dane 2881	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
180	179	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
181		resttopon 20965	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
182	133, 16, 181	mp2an 708	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
183		cncnp 21084	. . . 4 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))))
184	182, 133, 183	mp2an 708	. . 3 ⊢ (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦)))
185	7, 180, 184	sylanbrc 698	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
186		eqid 2622	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
187	41, 186, 137	cncfcn 22712	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
188	16, 130, 187	mp2an 708	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
189	185, 188	syl6eleqr 2712	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086  {csn 4177  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   “ cima 5117   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  supcsup 8346  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  +∞cpnf 10071  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  2c2 11070  ℕ₀cn0 11292  ℝ⁺crp 11832  [,)cico 12177  [,]cicc 12178  seqcseq 12801  ↑cexp 12860  abscabs 13974   ⇝ cli 14215  Σcsu 14416   ↾_t crest 16081  TopOpenctopn 16082  ∞Metcxmt 19731  ballcbl 19733  MetOpencmopn 19736  ℂ_fldccnfld 19746  Topctop 20698  TopOnctopon 20715  Clsdccld 20820  intcnt 20821   Cn ccn 21028   CnP ccnp 21029  Hauscha 21112  –cn→ccncf 22679

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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ulm 24131

This theorem is referenced by:  abelth2  24196