Theorem abelth 26419

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 26404.) (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 26412	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
8	1, 2, 3, 4, 5, 6	abelthlem9 26418	. . . . . . . . . 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 26410	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
10	9	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ 𝑆)
11	10	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 1 ∈ 𝑆)
12		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
13	11, 12	ovresd 7527	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14		ax-1cn 11087	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
15	5	ssrab3 4023	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 ⊆ ℂ
16	15, 12	sselid 3920	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
17		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24745	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
19	14, 16, 18	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
20	13, 19	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
21	20	breq1d 5096	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
22	7	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
23	22, 11	ffvelcdmd 7031	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) ∈ ℂ)
24	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
25	24	ffvelcdmda 7030	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
26	17	cnmetdval 24745	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘1) ∈ ℂ ∧ (𝐹‘𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
27	23, 25, 26	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
28	27	breq1d 5096	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟))
29	21, 28	imbi12d 344	. . . . . . . . . . . 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)))
30	29	ralbidva 3159	. . . . . . . . . . 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)))
31	30	rexbidv 3162	. . . . . . . . . 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)))
32	8, 31	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
33	32	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
34		cnxmet 24747	. . . . . . . . . 10 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
35		xmetres2 24336	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
36	34, 15, 35	mp2an 693	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
37		eqid 2737	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
38		eqid 2737	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
39	38	cnfldtopn 24756	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
40		eqid 2737	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
41	37, 39, 40	metrest 24499	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
42	34, 15, 41	mp2an 693	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
43	42, 39	metcnp 24516	. . . . . . . . 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))))
44	36, 34, 10, 43	mp3an12i 1468	. . . . . . . 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))))
45	7, 33, 44	mpbir2and 714	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
46	45	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
47		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
48	47	fveq2d 6838	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
49	46, 48	eleqtrrd 2840	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
50		eldifsn 4730	. . . . . . 7 ⊢ (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1))
51	9	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
52		abscl 15231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
53	52	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ)
54	53	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) ∈ ℝ))
55		absge0 15240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → 0 ≤ (abs‘𝑤))
56	55	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 0 ≤ (abs‘𝑤))
57	56	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → 0 ≤ (abs‘𝑤)))
58	1, 2	abelthlem1 26409	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
59	58	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
60	53	rexrd 11186	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ*)
61		1re 11135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
62		rexr 11182	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*)
63	61, 62	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ∈ ℝ*)
64		iccssxr 13374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0[,]+∞) ⊆ ℝ*
65		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))
66		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . 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*, < )
67	65, 1, 66	radcnvcl 26395	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
68	64, 67	sselid 3920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
69	68	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
70		xrltletr 13099	. . . . . . . . . . . . . . . . . . . . . . 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*, < )))
71	60, 63, 69, 70	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . 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*, < )))
72	59, 71	mpan2d 695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
73	54, 57, 72	3jcad 1130	. . . . . . . . . . . . . . . . . . . 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*, < ))))
74		0cn 11127	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
75	17	cnmetdval 24745	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
76	74, 75	mpan 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
77		abssub 15280	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
78	74, 77	mpan 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
79		subid1 11405	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
80	79	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
81	76, 78, 80	3eqtrd 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
82	81	breq1d 5096	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
83	82	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
84		0re 11137	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
85		elico2 13354	. . . . . . . . . . . . . . . . . . . . 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*, < ))))
86	84, 69, 85	sylancr 588	. . . . . . . . . . . . . . . . . . . 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*, < ))))
87	73, 83, 86	3imtr4d 294	. . . . . . . . . . . . . . . . . . 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*, < ))))
88	87	imdistanda 571	. . . . . . . . . . . . . . . . . 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*, < )))))
89		1xr 11195	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ*
90		elbl 24363	. . . . . . . . . . . . . . . . . . 19 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1)))
91	34, 74, 89, 90	mp3an 1464	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
92		absf 15291	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
93		ffn 6662	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
94		elpreima 7004	. . . . . . . . . . . . . . . . . . 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*, < )))))
95	92, 93, 94	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*, < ))))
96	88, 91, 95	3imtr4g 296	. . . . . . . . . . . . . . . . 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*, < )))))
97	96	ssrdv 3928	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0(ball‘(abs ∘ − ))1) ⊆ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
98	51, 97	sstrd 3933	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
99	98	resmptd 5999	. . . . . . . . . . . . . 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))))
100	6	reseq1i 5934	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1}))
101		difss 4077	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∖ {1}) ⊆ 𝑆
102		resmpt 5996	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∖ {1}) ⊆ 𝑆 → ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))))
103	101, 102	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
104	100, 103	eqtri 2760	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
105	99, 104	eqtr4di 2790	. . . . . . . . . . . . 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})))
106		cnvimass 6041	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
107	92	fdmi 6673	. . . . . . . . . . . . . . . . . . 19 ⊢ dom abs = ℂ
108	106, 107	sseqtri 3971	. . . . . . . . . . . . . . . . . 18 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
109	108	sseli 3918	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → 𝑥 ∈ ℂ)
110		fveq2 6834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗))
111		oveq2 7368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑥↑𝑛) = (𝑥↑𝑗))
112	110, 111	oveq12d 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑗) · (𝑥↑𝑗)))
113	112	cbvsumv 15649	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ0 ((𝐴‘𝑗) · (𝑥↑𝑗))
114	65	pserval2 26389	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗) = ((𝐴‘𝑗) · (𝑥↑𝑗)))
115	114	sumeq2dv 15655	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗) = Σ𝑗 ∈ ℕ0 ((𝐴‘𝑗) · (𝑥↑𝑗)))
116	113, 115	eqtr4id 2791	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗))
117	109, 116	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))
118	117	mpteq2ia 5181	. . . . . . . . . . . . . . 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))
119		eqid 2737	. . . . . . . . . . . . . . 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*, < )))
120		eqid 2737	. . . . . . . . . . . . . . 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))
121	65, 118, 1, 66, 119, 120	psercn 26404	. . . . . . . . . . . . . 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))
122		rescncf 24874	. . . . . . . . . . . . . 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)))
123	98, 121, 122	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))
124	105, 123	eqeltrrd 2838	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
125	124	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
126	101, 15	sstri 3932	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ {1}) ⊆ ℂ
127		ssid 3945	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
128		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
129	38	cnfldtopon 24757	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
130	129	toponrestid 22896	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
131	38, 128, 130	cncfcn 24887	. . . . . . . . . . . 12 ⊢ (((𝑆 ∖ {1}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
132	126, 127, 131	mp2an 693	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld))
133	125, 132	eleqtrdi 2847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
134		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
135		resttopon 23136	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑆 ∖ {1}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1})))
136	129, 126, 135	mp2an 693	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1}))
137	136	toponunii 22891	. . . . . . . . . . 11 ⊢ (𝑆 ∖ {1}) = ∪ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
138	137	cncnpi 23253	. . . . . . . . . 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))
139	133, 134, 138	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
140	38	cnfldtop 24758	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) ∈ Top
141		cnex 11110	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
142	141, 15	ssexi 5259	. . . . . . . . . . . 12 ⊢ 𝑆 ∈ V
143		restabs 23140	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆 ∧ 𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})))
144	140, 101, 142, 143	mp3an 1464	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
145	144	oveq1i 7370	. . . . . . . . . 10 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))
146	145	fveq1i 6835	. . . . . . . . 9 ⊢ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)
147	139, 146	eleqtrrdi 2848	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
148		resttop 23135	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
149	140, 142, 148	mp2an 693	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
150	149	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
151	101	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝑆 ∖ {1}) ⊆ 𝑆)
152	10	snssd 4753	. . . . . . . . . . . . 13 ⊢ (𝜑 → {1} ⊆ 𝑆)
153	38	cnfldhaus 24759	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂfld) ∈ Haus
154		unicntop 24760	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
155	154	sncld 23346	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂfld)))
156	153, 14, 155	mp2an 693	. . . . . . . . . . . . . 14 ⊢ {1} ∈ (Clsd‘(TopOpen‘ℂfld))
157	154	restcldi 23148	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
158	15, 156, 157	mp3an12 1454	. . . . . . . . . . . . 13 ⊢ ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
159	154	restuni 23137	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
160	140, 15, 159	mp2an 693	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
161	160	cldopn 23006	. . . . . . . . . . . . 13 ⊢ ({1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
162	152, 158, 161	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
163	160	isopn3 23041	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) → ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1})))
164	149, 101, 163	mp2an 693	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
165	162, 164	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
166	165	eleq2d 2823	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
167	166	biimpar 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})))
168	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
169	160, 154	cnprest 23264	. . . . . . . . 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)))
170	150, 151, 167, 168, 169	syl22anc 839	. . . . . . . 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)))
171	147, 170	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
172	50, 171	sylan2br 596	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
173	172	anassrs 467	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
174	49, 173	pm2.61dane 3020	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
175	174	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
176		resttopon 23136	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
177	129, 15, 176	mp2an 693	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
178		cncnp 23255	. . . 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))))
179	177, 129, 178	mp2an 693	. . 3 ⊢ (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦)))
180	7, 175, 179	sylanbrc 584	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
181		eqid 2737	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
182	38, 181, 130	cncfcn 24887	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
183	15, 127, 182	mp2an 693	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
184	180, 183	eleqtrrdi 2848	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ifcif 4467  {csn 4568  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   “ cima 5627   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  supcsup 9346  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034  +∞cpnf 11167  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171   − cmin 11368   / cdiv 11798  2c2 12227  ℕ₀cn0 12428  ℝ⁺crp 12933  [,)cico 13291  [,]cicc 13292  seqcseq 13954  ↑cexp 14014  abscabs 15187   ⇝ cli 15437  Σcsu 15639   ↾_t crest 17374  TopOpenctopn 17375  ∞Metcxmet 21329  ballcbl 21331  MetOpencmopn 21334  ℂ_fldccnfld 21344  Topctop 22868  TopOnctopon 22885  Clsdccld 22991  intcnt 22992   Cn ccn 23199   CnP ccnp 23200  Hauscha 23283  –cn→ccncf 24853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-addf 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-fi 9317  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-hash 14284  df-shft 15020  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15424  df-clim 15441  df-rlim 15442  df-sum 15640  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-rest 17376  df-topn 17377  df-0g 17395  df-gsum 17396  df-topgen 17397  df-pt 17398  df-prds 17401  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-mulg 19035  df-cntz 19283  df-cmn 19748  df-psmet 21336  df-xmet 21337  df-met 21338  df-bl 21339  df-mopn 21340  df-cnfld 21345  df-top 22869  df-topon 22886  df-topsp 22908  df-bases 22921  df-cld 22994  df-ntr 22995  df-cn 23202  df-cnp 23203  df-t1 23289  df-haus 23290  df-tx 23537  df-hmeo 23730  df-xms 24295  df-ms 24296  df-tms 24297  df-cncf 24855  df-ulm 26355

This theorem is referenced by:  abelth2  26420