Theorem abelth 26677

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 26662.) (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 26670	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
8	1, 2, 3, 4, 5, 6	abelthlem9 26676	. . . . . . . . . 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 26668	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
10	9	simpld 500	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ 𝑆)
11	10	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 1 ∈ 𝑆)
12		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
13	11, 12	ovresd 7580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14		ax-1cn 11182	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
15	5	ssrab3 4030	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 ⊆ ℂ
16	15, 12	sselid 3929	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
17		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24996	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
19	14, 16, 18	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
20	13, 19	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
21	20	breq1d 5113	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
22	7	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
23	22, 11	ffvelcdmd 7078	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) ∈ ℂ)
24	7	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
25	24	ffvelcdmda 7077	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
26	17	cnmetdval 24996	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘1) ∈ ℂ ∧ (𝐹‘𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
27	23, 25, 26	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
28	27	breq1d 5113	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑦 ∈ 𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟))
29	21, 28	imbi12d 347	. . . . . . . . . . . 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 3183	. . . . . . . . . . 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 3186	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
33	32	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
34		cnxmet 24998	. . . . . . . . . 10 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
35		xmetres2 24587	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
36	34, 15, 35	mp2an 705	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
37		eqid 2760	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
38		eqid 2760	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
39	38	cnfldtopn 25007	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
40		eqid 2760	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
41	37, 39, 40	metrest 24750	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
42	34, 15, 41	mp2an 705	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
43	42, 39	metcnp 24767	. . . . . . . . 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 1494	. . . . . . . 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 726	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
46	45	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
47		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
48	47	fveq2d 6882	. . . . . 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 2863	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
50		eldifsn 4748	. . . . . . 7 ⊢ (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1))
51	9	simprd 501	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
52		abscl 15365	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
53	52	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ)
54	53	a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) ∈ ℝ))
55		absge0 15374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → 0 ≤ (abs‘𝑤))
56	55	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 0 ≤ (abs‘𝑤))
57	56	a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → 0 ≤ (abs‘𝑤)))
58	1, 2	abelthlem1 26667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
59	58	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
60	53	rexrd 11283	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ*)
61		1re 11232	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
62		rexr 11279	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℝ → 1 ∈ ℝ*)
63	61, 62	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ∈ ℝ*)
64		iccssxr 13483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0[,]+∞) ⊆ ℝ*
65		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))
66		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . . 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 26653	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
68	64, 67	sselid 3929	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
69	68	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
70		xrltletr 13208	. . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . 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 707	. . . . . . . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . . . . . . 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 11222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
75	17	cnmetdval 24996	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
76	74, 75	mpan 703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
77		abssub 15414	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
78	74, 77	mpan 703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
79		subid1 11502	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
80	79	fveq2d 6882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
81	76, 78, 80	3eqtrd 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
82	81	breq1d 5113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
83	82	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
84		0re 11234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
85		elico2 13463	. . . . . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . . . . 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 297	. . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . 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 11292	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ*
90		elbl 24614	. . . . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
92		absf 15425	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
93		ffn 6702	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
94		elpreima 7050	. . . . . . . . . . . . . . . . . . 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 299	. . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
99	98	resmptd 6036	. . . . . . . . . . . . . 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 5968	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1}))
101		difss 4083	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∖ {1}) ⊆ 𝑆
102		resmpt 6033	. . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)))
105	99, 104	eqtr4di 2813	. . . . . . . . . . . . 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 6078	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
107	92	fdmi 6714	. . . . . . . . . . . . . . . . . . 19 ⊢ dom abs = ℂ
108	106, 107	sseqtri 3979	. . . . . . . . . . . . . . . . . 18 ⊢ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
109	108	sseli 3927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → 𝑥 ∈ ℂ)
110		fveq2 6878	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝐴‘𝑛) = (𝐴‘𝑗))
111		oveq2 7421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → (𝑥↑𝑛) = (𝑥↑𝑗))
112	110, 111	oveq12d 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑗) · (𝑥↑𝑗)))
113	112	cbvsumv 15783	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑛 ∈ ℕ0 ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ0 ((𝐴‘𝑗) · (𝑥↑𝑗))
114	65	pserval2 26647	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑥)‘𝑗) = ((𝐴‘𝑗) · (𝑥↑𝑗)))
115	114	sumeq2dv 15789	. . . . . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 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 5200	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 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 26662	. . . . . . . . . . . . . 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 25125	. . . . . . . . . . . . . 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 66	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
125	124	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
126	101, 15	sstri 3940	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ {1}) ⊆ ℂ
127		ssid 3953	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
128		eqid 2760	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
129	38	cnfldtopon 25008	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
130	129	toponrestid 23146	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
131	38, 128, 130	cncfcn 25138	. . . . . . . . . . . 12 ⊢ (((𝑆 ∖ {1}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
132	126, 127, 131	mp2an 705	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld))
133	125, 132	eleqtrdi 2870	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
134		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
135		resttopon 23386	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑆 ∖ {1}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1})))
136	129, 126, 135	mp2an 705	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1}))
137	136	toponunii 23141	. . . . . . . . . . 11 ⊢ (𝑆 ∖ {1}) = ∪ ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
138	137	cncnpi 23503	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
140	38	cnfldtop 25009	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) ∈ Top
141		cnex 11205	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
142	141, 15	ssexi 5287	. . . . . . . . . . . 12 ⊢ 𝑆 ∈ V
143		restabs 23390	. . . . . . . . . . . 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 1490	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
145	144	oveq1i 7423	. . . . . . . . . 10 ⊢ ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))
146	145	fveq1i 6879	. . . . . . . . 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 2871	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
148		resttop 23385	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
149	140, 142, 148	mp2an 705	. . . . . . . . . 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 4747	. . . . . . . . . . . . 13 ⊢ (𝜑 → {1} ⊆ 𝑆)
153	38	cnfldhaus 25010	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂfld) ∈ Haus
154		unicntop 25011	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂfld)
155	154	sncld 23596	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂfld)))
156	153, 14, 155	mp2an 705	. . . . . . . . . . . . . 14 ⊢ {1} ∈ (Clsd‘(TopOpen‘ℂfld))
157	154	restcldi 23398	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
158	15, 156, 157	mp3an12 1480	. . . . . . . . . . . . 13 ⊢ ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
159	154	restuni 23387	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆))
160	140, 15, 159	mp2an 705	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ∪ ((TopOpen‘ℂfld) ↾t 𝑆)
161	160	cldopn 23256	. . . . . . . . . . . . 13 ⊢ ({1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
162	152, 158, 161	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
163	160	isopn3 23291	. . . . . . . . . . . . 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 705	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
165	162, 164	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
166	165	eleq2d 2846	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
167	166	biimpar 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})))
168	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
169	160, 154	cnprest 23514	. . . . . . . . 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 852	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
172	50, 171	sylan2br 607	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
173	172	anassrs 473	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
174	49, 173	pm2.61dane 3042	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
175	174	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
176		resttopon 23386	. . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
177	129, 15, 176	mp2an 705	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
178		cncnp 23505	. . . 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 705	. . 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 595	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
181		eqid 2760	. . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
182	38, 181, 130	cncfcn 25138	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
183	15, 127, 182	mp2an 705	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
184	180, 183	eleqtrrdi 2871	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  ifcif 4482  {csn 4584  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  dom cdm 5655   ↾ cres 5657   “ cima 5658   ∘ ccom 5659   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  supcsup 9410  ℂcc 11122  ℝcr 11123  0cc0 11124  1c1 11125   + caddc 11127   · cmul 11129  +∞cpnf 11264  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268   − cmin 11465   / cdiv 11895  2c2 12319  ℕ₀cn0 12528  ℝ⁺crp 13042  [,)cico 13400  [,]cicc 13401  seqcseq 14065  ↑cexp 14125  abscabs 15321   ⇝ cli 15571  Σcsu 15773   ↾_t crest 17505  TopOpenctopn 17506  ∞Metcxmet 21570  ballcbl 21572  MetOpencmopn 21575  ℂ_fldccnfld 21585  Topctop 23118  TopOnctopon 23135  Clsdccld 23241  intcnt 23242   Cn ccn 23449   CnP ccnp 23450  Hauscha 23533  –cn→ccncf 25104

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-addf 11203

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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-tp 4589  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-dec 12737  df-uz 12888  df-q 12998  df-rp 13043  df-xneg 13163  df-xadd 13164  df-xmul 13165  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-seq 14066  df-exp 14126  df-hash 14395  df-shft 15140  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-limsup 15558  df-clim 15575  df-rlim 15576  df-sum 15774  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-starv 17357  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-unif 17365  df-hom 17366  df-cco 17367  df-rest 17507  df-topn 17508  df-0g 17526  df-gsum 17527  df-topgen 17528  df-pt 17529  df-prds 17532  df-xrs 17588  df-qtop 17593  df-imas 17594  df-xps 17596  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-submnd 18892  df-mulg 19191  df-cntz 19444  df-cmn 19909  df-psmet 21577  df-xmet 21578  df-met 21579  df-bl 21580  df-mopn 21581  df-cnfld 21586  df-top 23119  df-topon 23136  df-topsp 23158  df-bases 23171  df-cld 23244  df-ntr 23245  df-cn 23452  df-cnp 23453  df-t1 23539  df-haus 23540  df-tx 23788  df-hmeo 23981  df-xms 24546  df-ms 24547  df-tms 24548  df-cncf 25106  df-ulm 26613

This theorem is used by:  abelth2  26678