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Theorem abelth 26423
Description: Abel's theorem. If the power series Σ𝑛 ∈ ℕ0𝐴(𝑛)(𝑥𝑛) 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 26408.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
Hypotheses
Ref Expression
abelth.1 (𝜑𝐴:ℕ0⟶ℂ)
abelth.2 (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
abelth.3 (𝜑𝑀 ∈ ℝ)
abelth.4 (𝜑 → 0 ≤ 𝑀)
abelth.5 𝑆 = {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))}
abelth.6 𝐹 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
Assertion
Ref Expression
abelth (𝜑𝐹 ∈ (𝑆cn→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.
StepHypRef Expression
1 abelth.1 . . . 4 (𝜑𝐴:ℕ0⟶ℂ)
2 abelth.2 . . . 4 (𝜑 → seq0( + , 𝐴) ∈ dom ⇝ )
3 abelth.3 . . . 4 (𝜑𝑀 ∈ 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 = (𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
71, 2, 3, 4, 5, 6abelthlem4 26416 . . 3 (𝜑𝐹:𝑆⟶ℂ)
81, 2, 3, 4, 5, 6abelthlem9 26422 . . . . . . . . . 10 ((𝜑𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟))
91, 2, 3, 4, 5abelthlem2 26414 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
109simpld 493 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ 𝑆)
1110ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 1 ∈ 𝑆)
12 simpr 483 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝑦𝑆)
1311, 12ovresd 7588 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14 ax-1cn 11198 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
155ssrab3 4076 . . . . . . . . . . . . . . . . 17 𝑆 ⊆ ℂ
1615, 12sselid 3974 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝑦 ∈ ℂ)
17 eqid 2725 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) = (abs ∘ − )
1817cnmetdval 24731 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
1914, 16, 18sylancr 585 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
2013, 19eqtrd 2765 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
2120breq1d 5159 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
227ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝐹:𝑆⟶ℂ)
2322, 11ffvelcdmd 7094 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (𝐹‘1) ∈ ℂ)
247adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
2524ffvelcdmda 7093 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (𝐹𝑦) ∈ ℂ)
2617cnmetdval 24731 . . . . . . . . . . . . . . 15 (((𝐹‘1) ∈ ℂ ∧ (𝐹𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) = (abs‘((𝐹‘1) − (𝐹𝑦))))
2723, 25, 26syl2anc 582 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) = (abs‘((𝐹‘1) − (𝐹𝑦))))
2827breq1d 5159 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟))
2921, 28imbi12d 343 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟) ↔ ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟)))
3029ralbidva 3165 . . . . . . . . . . 11 ((𝜑𝑟 ∈ ℝ+) → (∀𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟) ↔ ∀𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟)))
3130rexbidv 3168 . . . . . . . . . 10 ((𝜑𝑟 ∈ ℝ+) → (∃𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟) ↔ ∃𝑤 ∈ ℝ+𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟)))
328, 31mpbird 256 . . . . . . . . 9 ((𝜑𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))
3332ralrimiva 3135 . . . . . . . 8 (𝜑 → ∀𝑟 ∈ ℝ+𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))
34 cnxmet 24733 . . . . . . . . . 10 (abs ∘ − ) ∈ (∞Met‘ℂ)
35 xmetres2 24311 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
3634, 15, 35mp2an 690 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
37 eqid 2725 . . . . . . . . . . . 12 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
38 eqid 2725 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
3938cnfldtopn 24742 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
40 eqid 2725 . . . . . . . . . . . 12 (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
4137, 39, 40metrest 24477 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
4234, 15, 41mp2an 690 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
4342, 39metcnp 24494 . . . . . . . . 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+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))))
4436, 34, 10, 43mp3an12i 1461 . . . . . . . 8 (𝜑 → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑟 ∈ ℝ+𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))))
457, 33, 44mpbir2and 711 . . . . . . 7 (𝜑𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
4645ad2antrr 724 . . . . . 6 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
47 simpr 483 . . . . . . 7 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
4847fveq2d 6900 . . . . . 6 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
4946, 48eleqtrrd 2828 . . . . 5 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
50 eldifsn 4792 . . . . . . 7 (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦𝑆𝑦 ≠ 1))
519simprd 494 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
52 abscl 15261 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
5352adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ)
5453a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) ∈ ℝ))
55 absge0 15270 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → 0 ≤ (abs‘𝑤))
5655adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → 0 ≤ (abs‘𝑤))
5756a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → 0 ≤ (abs‘𝑤)))
581, 2abelthlem1 26413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
5958adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
6053rexrd 11296 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ*)
61 1re 11246 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℝ
62 rexr 11292 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ → 1 ∈ ℝ*)
6361, 62mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → 1 ∈ ℝ*)
64 iccssxr 13442 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0[,]+∞) ⊆ ℝ*
65 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))
66 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) = sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )
6765, 1, 66radcnvcl 26398 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
6864, 67sselid 3974 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
6968adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
70 xrltletr 13171 . . . . . . . . . . . . . . . . . . . . . . 23 (((abs‘𝑤) ∈ ℝ* ∧ 1 ∈ ℝ* ∧ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) → (((abs‘𝑤) < 1 ∧ 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
7160, 63, 69, 70syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → (((abs‘𝑤) < 1 ∧ 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
7259, 71mpan2d 692 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
7354, 57, 723jcad 1126 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → ((abs‘𝑤) ∈ ℝ ∧ 0 ≤ (abs‘𝑤) ∧ (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
74 0cn 11238 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℂ
7517cnmetdval 24731 . . . . . . . . . . . . . . . . . . . . . . . 24 ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
7674, 75mpan 688 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
77 abssub 15309 . . . . . . . . . . . . . . . . . . . . . . . 24 ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
7874, 77mpan 688 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
79 subid1 11512 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
8079fveq2d 6900 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
8176, 78, 803eqtrd 2769 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
8281breq1d 5159 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
8382adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
84 0re 11248 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ
85 elico2 13423 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ ∧ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*) → ((abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) ↔ ((abs‘𝑤) ∈ ℝ ∧ 0 ≤ (abs‘𝑤) ∧ (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
8684, 69, 85sylancr 585 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) ↔ ((abs‘𝑤) ∈ ℝ ∧ 0 ≤ (abs‘𝑤) ∧ (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
8773, 83, 863imtr4d 293 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 → (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
8887imdistanda 570 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1) → (𝑤 ∈ ℂ ∧ (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
89 1xr 11305 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ*
90 elbl 24338 . . . . . . . . . . . . . . . . . . 19 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1)))
9134, 74, 89, 90mp3an 1457 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
92 absf 15320 . . . . . . . . . . . . . . . . . . 19 abs:ℂ⟶ℝ
93 ffn 6723 . . . . . . . . . . . . . . . . . . 19 (abs:ℂ⟶ℝ → abs Fn ℂ)
94 elpreima 7066 . . . . . . . . . . . . . . . . . . 19 (abs Fn ℂ → (𝑤 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↔ (𝑤 ∈ ℂ ∧ (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
9592, 93, 94mp2b 10 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↔ (𝑤 ∈ ℂ ∧ (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
9688, 91, 953imtr4g 295 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) → 𝑤 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
9796ssrdv 3982 . . . . . . . . . . . . . . . 16 (𝜑 → (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
9851, 97sstrd 3987 . . . . . . . . . . . . . . 15 (𝜑 → (𝑆 ∖ {1}) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
9998resmptd 6045 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))))
1006reseq1i 5981 . . . . . . . . . . . . . . 15 (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1}))
101 difss 4128 . . . . . . . . . . . . . . . 16 (𝑆 ∖ {1}) ⊆ 𝑆
102 resmpt 6042 . . . . . . . . . . . . . . . 16 ((𝑆 ∖ {1}) ⊆ 𝑆 → ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))))
103101, 102ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
104100, 103eqtri 2753 . . . . . . . . . . . . . 14 (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
10599, 104eqtr4di 2783 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) = (𝐹 ↾ (𝑆 ∖ {1})))
106 cnvimass 6086 . . . . . . . . . . . . . . . . . . 19 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
10792fdmi 6734 . . . . . . . . . . . . . . . . . . 19 dom abs = ℂ
108106, 107sseqtri 4013 . . . . . . . . . . . . . . . . . 18 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
109108sseli 3972 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → 𝑥 ∈ ℂ)
110 fveq2 6896 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
111 oveq2 7427 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝑥𝑛) = (𝑥𝑗))
112110, 111oveq12d 7437 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((𝐴𝑛) · (𝑥𝑛)) = ((𝐴𝑗) · (𝑥𝑗)))
113112cbvsumv 15678 . . . . . . . . . . . . . . . . . 18 Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)) = Σ𝑗 ∈ ℕ0 ((𝐴𝑗) · (𝑥𝑗))
11465pserval2 26392 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗) = ((𝐴𝑗) · (𝑥𝑗)))
115114sumeq2dv 15685 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℂ → Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗) = Σ𝑗 ∈ ℕ0 ((𝐴𝑗) · (𝑥𝑗)))
116113, 115eqtr4id 2784 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℂ → Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)) = Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗))
117109, 116syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)) = Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗))
118117mpteq2ia 5252 . . . . . . . . . . . . . . 15 (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) = (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗))
119 eqid 2725 . . . . . . . . . . . . . . 15 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) = (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
120 eqid 2725 . . . . . . . . . . . . . . 15 if(sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑣) + sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑣) + 1)) = if(sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ, (((abs‘𝑣) + sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )) / 2), ((abs‘𝑣) + 1))
12165, 118, 1, 66, 119, 120psercn 26408 . . . . . . . . . . . . . 14 (𝜑 → (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ∈ ((abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ))
122 rescncf 24861 . . . . . . . . . . . . . 14 ((𝑆 ∖ {1}) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → ((𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ∈ ((abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))–cn→ℂ) → ((𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ)))
12398, 121, 122sylc 65 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
124105, 123eqeltrrd 2826 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
125124adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
126101, 15sstri 3986 . . . . . . . . . . . 12 (𝑆 ∖ {1}) ⊆ ℂ
127 ssid 3999 . . . . . . . . . . . 12 ℂ ⊆ ℂ
128 eqid 2725 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
12938cnfldtopon 24743 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
130129toponrestid 22867 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
13138, 128, 130cncfcn 24874 . . . . . . . . . . . 12 (((𝑆 ∖ {1}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
132126, 127, 131mp2an 690 . . . . . . . . . . 11 ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld))
133125, 132eleqtrdi 2835 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
134 simpr 483 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
135 resttopon 23109 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑆 ∖ {1}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1})))
136129, 126, 135mp2an 690 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1}))
137136toponunii 22862 . . . . . . . . . . 11 (𝑆 ∖ {1}) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
138137cncnpi 23226 . . . . . . . . . 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))
139133, 134, 138syl2anc 582 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
14038cnfldtop 24744 . . . . . . . . . . . 12 (TopOpen‘ℂfld) ∈ Top
141 cnex 11221 . . . . . . . . . . . . 13 ℂ ∈ V
142141, 15ssexi 5323 . . . . . . . . . . . 12 𝑆 ∈ V
143 restabs 23113 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})))
144140, 101, 142, 143mp3an 1457 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
145144oveq1i 7429 . . . . . . . . . 10 ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))
146145fveq1i 6897 . . . . . . . . 9 (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)
147139, 146eleqtrrdi 2836 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
148 resttop 23108 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
149140, 142, 148mp2an 690 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
150149a1i 11 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
151101a1i 11 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝑆 ∖ {1}) ⊆ 𝑆)
15210snssd 4814 . . . . . . . . . . . . 13 (𝜑 → {1} ⊆ 𝑆)
15338cnfldhaus 24745 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ Haus
154 unicntop 24746 . . . . . . . . . . . . . . . 16 ℂ = (TopOpen‘ℂfld)
155154sncld 23319 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂfld)))
156153, 14, 155mp2an 690 . . . . . . . . . . . . . 14 {1} ∈ (Clsd‘(TopOpen‘ℂfld))
157154restcldi 23121 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
15815, 156, 157mp3an12 1447 . . . . . . . . . . . . 13 ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
159154restuni 23110 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
160140, 15, 159mp2an 690 . . . . . . . . . . . . . 14 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
161160cldopn 22979 . . . . . . . . . . . . 13 ({1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
162152, 158, 1613syl 18 . . . . . . . . . . . 12 (𝜑 → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
163160isopn3 23014 . . . . . . . . . . . . 13 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) → ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1})))
164149, 101, 163mp2an 690 . . . . . . . . . . . 12 ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
165162, 164sylib 217 . . . . . . . . . . 11 (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
166165eleq2d 2811 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
167166biimpar 476 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})))
1687adantr 479 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
169160, 154cnprest 23237 . . . . . . . . 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)))
170150, 151, 167, 168, 169syl22anc 837 . . . . . . . 8 ((𝜑𝑦 ∈ (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)))
171147, 170mpbird 256 . . . . . . 7 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
17250, 171sylan2br 593 . . . . . 6 ((𝜑 ∧ (𝑦𝑆𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
173172anassrs 466 . . . . 5 (((𝜑𝑦𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
17449, 173pm2.61dane 3018 . . . 4 ((𝜑𝑦𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
175174ralrimiva 3135 . . 3 (𝜑 → ∀𝑦𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
176 resttopon 23109 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
177129, 15, 176mp2an 690 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
178 cncnp 23228 . . . 4 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))))
179177, 129, 178mp2an 690 . . 3 (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦)))
1807, 175, 179sylanbrc 581 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
181 eqid 2725 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
18238, 181, 130cncfcn 24874 . . 3 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
18315, 127, 182mp2an 690 . 2 (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
184180, 183eleqtrrdi 2836 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  cdif 3941  wss 3944  ifcif 4530  {csn 4630   cuni 4909   class class class wbr 5149  cmpt 5232   × cxp 5676  ccnv 5677  dom cdm 5678  cres 5680  cima 5681  ccom 5682   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  supcsup 9465  cc 11138  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   · cmul 11145  +∞cpnf 11277  *cxr 11279   < clt 11280  cle 11281  cmin 11476   / cdiv 11903  2c2 12300  0cn0 12505  +crp 13009  [,)cico 13361  [,]cicc 13362  seqcseq 14002  cexp 14062  abscabs 15217  cli 15464  Σcsu 15668  t crest 17405  TopOpenctopn 17406  ∞Metcxmet 21281  ballcbl 21283  MetOpencmopn 21286  fldccnfld 21296  Topctop 22839  TopOnctopon 22856  Clsdccld 22964  intcnt 22965   Cn ccn 23172   CnP ccnp 23173  Hauscha 23256  cnccncf 24840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218  ax-addf 11219
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-fi 9436  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-shft 15050  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-limsup 15451  df-clim 15468  df-rlim 15469  df-sum 15669  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-hom 17260  df-cco 17261  df-rest 17407  df-topn 17408  df-0g 17426  df-gsum 17427  df-topgen 17428  df-pt 17429  df-prds 17432  df-xrs 17487  df-qtop 17492  df-imas 17493  df-xps 17495  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-mulg 19032  df-cntz 19280  df-cmn 19749  df-psmet 21288  df-xmet 21289  df-met 21290  df-bl 21291  df-mopn 21292  df-cnfld 21297  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cld 22967  df-ntr 22968  df-cn 23175  df-cnp 23176  df-t1 23262  df-haus 23263  df-tx 23510  df-hmeo 23703  df-xms 24270  df-ms 24271  df-tms 24272  df-cncf 24842  df-ulm 26358
This theorem is referenced by:  abelth2  26424
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