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Theorem abelth 26187

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 26172.) (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 26180	. . 3 ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)
8	1, 2, 3, 4, 5, 6	abelthlem9 26186	. . . . . . . . . 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 26178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
10	9	simpld 493	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ 𝑆)
11	10	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → 1 ∈ 𝑆)
12		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆)
13	11, 12	ovresd 7578	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14		ax-1cn 11172	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
15	5	ssrab3 4081	. . . . . . . . . . . . . . . . 17 ⊢ 𝑆 ⊆ ℂ
16	15, 12	sselid 3981	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
17		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) = (abs ∘ − )
18	17	cnmetdval 24509	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
19	14, 16, 18	sylancr 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
20	13, 19	eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
21	20	breq1d 5159	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
22	7	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → 𝐹:𝑆⟶ℂ)
23	22, 11	ffvelcdmd 7088	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (𝐹‘1) ∈ ℂ)
24	7	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝐹:𝑆⟶ℂ)
25	24	ffvelcdmda 7087	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (𝐹‘𝑦) ∈ ℂ)
26	17	cnmetdval 24509	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘1) ∈ ℂ ∧ (𝐹‘𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
27	23, 25, 26	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) = (abs‘((𝐹‘1) − (𝐹‘𝑦))))
28	27	breq1d 5159	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹‘𝑦))) < 𝑟))
29	21, 28	imbi12d 343	. . . . . . . . . . . 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 3173	. . . . . . . . . . 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 3176	. . . . . . . . . 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 256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑤 ∈ ℝ⁺ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
33	32	ralrimiva 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑟 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑦 ∈ 𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹‘𝑦)) < 𝑟))
34		cnxmet 24511	. . . . . . . . . 10 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
35		xmetres2 24089	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
36	34, 15, 35	mp2an 688	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
37		eqid 2730	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
38		eqid 2730	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
39	38	cnfldtopn 24520	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
40		eqid 2730	. . . . . . . . . . . 12 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
41	37, 39, 40	metrest 24255	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
42	34, 15, 41	mp2an 688	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
43	42, 39	metcnp 24272	. . . . . . . . 9 ⊢ ((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ (abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ 𝑆) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘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 1463	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘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 709	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘1))
46	45	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘1))
47		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
48	47	fveq2d 6896	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦) = ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘1))
49	46, 48	eleqtrrd 2834	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))
50		eldifsn 4791	. . . . . . 7 ⊢ (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1))
51	9	simprd 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
52		abscl 15231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
53	52	adantl 480	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 26177	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ))
59	58	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ))
60	53	rexrd 11270	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ^*)
61		1re 11220	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℝ
62		rexr 11266	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ ℝ → 1 ∈ ℝ^*)
63	61, 62	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → 1 ∈ ℝ^*)
64		iccssxr 13413	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0[,]+∞) ⊆ ℝ^*
65		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))
66		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . 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 26163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ) ∈ (0[,]+∞))
68	64, 67	sselid 3981	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
69	68	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^, < ) ∈ ℝ^)
70		xrltletr 13142	. . . . . . . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . . 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 11212	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ ℂ
75	17	cnmetdval 24509	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
76	74, 75	mpan 686	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
77		abssub 15279	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
78	74, 77	mpan 686	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
79		subid1 11486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
80	79	fveq2d 6896	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
81	76, 78, 80	3eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
82	81	breq1d 5159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
83	82	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
84		0re 11222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ ℝ
85		elico2 13394	. . . . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . . . . . . 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 570	. . . . . . . . . . . . . . . . . 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 11279	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
90		elbl 24116	. . . . . . . . . . . . . . . . . . 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 1459	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
92		absf 15290	. . . . . . . . . . . . . . . . . . 19 ⊢ abs:ℂ⟶ℝ
93		ffn 6718	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ)
94		elpreima 7060	. . . . . . . . . . . . . . . . . . 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 295	. . . . . . . . . . . . . . . . 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 3989	. . . . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 ∖ {1}) ⊆ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ))))
99	98	resmptd 6041	. . . . . . . . . . . . . 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 5978	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥 ∈ 𝑆 ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛))) ↾ (𝑆 ∖ {1}))
101		difss 4132	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∖ {1}) ⊆ 𝑆
102		resmpt 6038	. . . . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . 14 ⊢ (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)))
105	99, 104	eqtr4di 2788	. . . . . . . . . . . . 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 6081	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ dom abs
107	92	fdmi 6730	. . . . . . . . . . . . . . . . . . 19 ⊢ dom abs = ℂ
108	106, 107	sseqtri 4019	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ))) ⊆ ℂ
109	108	sseli 3979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (^◡abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐴‘𝑛) · (𝑡↑𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ^*, < ))) → 𝑥 ∈ ℂ)
110		fveq2 6892	. . . . . . . . . . . . . . . . . . . 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 15648	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑛 ∈ ℕ₀ ((𝐴‘𝑛) · (𝑥↑𝑛)) = Σ𝑗 ∈ ℕ₀ ((𝐴‘𝑗) · (𝑥↑𝑗))
114	65	pserval2 26157	. . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 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 5252	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . 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 26172	. . . . . . . . . . . . . 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 24639	. . . . . . . . . . . . . 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 2832	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
125	124	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
126	101, 15	sstri 3992	. . . . . . . . . . . 12 ⊢ (𝑆 ∖ {1}) ⊆ ℂ
127		ssid 4005	. . . . . . . . . . . 12 ⊢ ℂ ⊆ ℂ
128		eqid 2730	. . . . . . . . . . . . 13 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) = ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1}))
129	38	cnfldtopon 24521	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
130	129	toponrestid 22645	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
131	38, 128, 130	cncfcn 24652	. . . . . . . . . . . 12 ⊢ (((𝑆 ∖ {1}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) Cn (TopOpen‘ℂ_fld)))
132	126, 127, 131	mp2an 688	. . . . . . . . . . 11 ⊢ ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) Cn (TopOpen‘ℂ_fld))
133	125, 132	eleqtrdi 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) Cn (TopOpen‘ℂ_fld)))
134		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
135		resttopon 22887	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝑆 ∖ {1}) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1})))
136	129, 126, 135	mp2an 688	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1}))
137	136	toponunii 22640	. . . . . . . . . . 11 ⊢ (𝑆 ∖ {1}) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1}))
138	137	cncnpi 23004	. . . . . . . . . 10 ⊢ (((𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) Cn (TopOpen‘ℂ_fld)) ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦))
139	133, 134, 138	syl2anc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦))
140	38	cnfldtop 24522	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ Top
141		cnex 11195	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
142	141, 15	ssexi 5323	. . . . . . . . . . . 12 ⊢ 𝑆 ∈ V
143		restabs 22891	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆 ∧ 𝑆 ∈ V) → (((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) = ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})))
144	140, 101, 142, 143	mp3an 1459	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) = ((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1}))
145	144	oveq1i 7423	. . . . . . . . . 10 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld)) = (((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))
146	145	fveq1i 6893	. . . . . . . . 9 ⊢ (((((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦) = ((((TopOpen‘ℂ_fld) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦)
147	139, 146	eleqtrrdi 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦))
148		resttop 22886	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
149	140, 142, 148	mp2an 688	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top
150	149	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
151	101	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝑆 ∖ {1}) ⊆ 𝑆)
152	10	snssd 4813	. . . . . . . . . . . . 13 ⊢ (𝜑 → {1} ⊆ 𝑆)
153	38	cnfldhaus 24523	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) ∈ Haus
154		unicntop 24524	. . . . . . . . . . . . . . . 16 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
155	154	sncld 23097	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂ_fld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂ_fld)))
156	153, 14, 155	mp2an 688	. . . . . . . . . . . . . 14 ⊢ {1} ∈ (Clsd‘(TopOpen‘ℂ_fld))
157	154	restcldi 22899	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂ_fld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t 𝑆)))
158	15, 156, 157	mp3an12 1449	. . . . . . . . . . . . 13 ⊢ ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t 𝑆)))
159	154	restuni 22888	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
160	140, 15, 159	mp2an 688	. . . . . . . . . . . . . 14 ⊢ 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
161	160	cldopn 22757	. . . . . . . . . . . . 13 ⊢ ({1} ∈ (Clsd‘((TopOpen‘ℂ_fld) ↾_t 𝑆)) → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
162	152, 158, 161	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
163	160	isopn3 22792	. . . . . . . . . . . . 13 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) → ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ↔ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1})))
164	149, 101, 163	mp2an 688	. . . . . . . . . . . 12 ⊢ ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ↔ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
165	162, 164	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
166	165	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
167	166	biimpar 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘(𝑆 ∖ {1})))
168	7	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
169	160, 154	cnprest 23015	. . . . . . . . 9 ⊢ (((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) ∧ (𝑦 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘(𝑆 ∖ {1})) ∧ 𝐹:𝑆⟶ℂ)) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦)))
170	150, 151, 167, 168, 169	syl22anc 835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦) ↔ (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂ_fld) ↾_t 𝑆) ↾_t (𝑆 ∖ {1})) CnP (TopOpen‘ℂ_fld))‘𝑦)))
171	147, 170	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))
172	50, 171	sylan2br 593	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑆 ∧ 𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))
173	172	anassrs 466	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))
174	49, 173	pm2.61dane 3027	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))
175	174	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))
176		resttopon 22887	. . . . 5 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
177	129, 15, 176	mp2an 688	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆)
178		cncnp 23006	. . . 4 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦))))
179	177, 129, 178	mp2an 688	. . 3 ⊢ (𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦 ∈ 𝑆 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) CnP (TopOpen‘ℂ_fld))‘𝑦)))
180	7, 175, 179	sylanbrc 581	. 2 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)))
181		eqid 2730	. . . 4 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
182	38, 181, 130	cncfcn 24652	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld)))
183	15, 127, 182	mp2an 688	. 2 ⊢ (𝑆–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t 𝑆) Cn (TopOpen‘ℂ_fld))
184	180, 183	eleqtrrdi 2842	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3430 Vcvv 3472 ∖ cdif 3946 ⊆ wss 3949 ifcif 4529 {csn 4629 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ^◡ccnv 5676 dom cdm 5677 ↾ cres 5679 “ cima 5680 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 supcsup 9439 ℂcc 11112 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 · cmul 11119 +∞cpnf 11251 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 − cmin 11450 / cdiv 11877 2c2 12273 ℕ₀cn0 12478 ℝ⁺crp 12980 [,)cico 13332 [,]cicc 13333 seqcseq 13972 ↑cexp 14033 abscabs 15187 ⇝ cli 15434 Σcsu 15638 ↾_t crest 17372 TopOpenctopn 17373 ∞Metcxmet 21131 ballcbl 21133 MetOpencmopn 21136 ℂ_fldccnfld 21146 Topctop 22617 TopOnctopon 22634 Clsdccld 22742 intcnt 22743 Cn ccn 22950 CnP ccnp 22951 Hauscha 23034 –cn→ccncf 24618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14034 df-hash 14297 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18708 df-mulg 18989 df-cntz 19224 df-cmn 19693 df-psmet 21138 df-xmet 21139 df-met 21140 df-bl 21141 df-mopn 21142 df-cnfld 21147 df-top 22618 df-topon 22635 df-topsp 22657 df-bases 22671 df-cld 22745 df-ntr 22746 df-cn 22953 df-cnp 22954 df-t1 23040 df-haus 23041 df-tx 23288 df-hmeo 23481 df-xms 24048 df-ms 24049 df-tms 24050 df-cncf 24620 df-ulm 26123

This theorem is referenced by: abelth2 26188

W3C validator