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Theorem abelth 25021
 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 25006.) (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 25014 . . 3 (𝜑𝐹:𝑆⟶ℂ)
81, 2, 3, 4, 5, 6abelthlem9 25020 . . . . . . . . . 10 ((𝜑𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟))
91, 2, 3, 4, 5abelthlem2 25012 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
109simpld 497 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ 𝑆)
1110ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 1 ∈ 𝑆)
12 simpr 487 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝑦𝑆)
1311, 12ovresd 7307 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14 ax-1cn 10587 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
155ssrab3 4055 . . . . . . . . . . . . . . . . 17 𝑆 ⊆ ℂ
1615, 12sseldi 3963 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝑦 ∈ ℂ)
17 eqid 2819 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) = (abs ∘ − )
1817cnmetdval 23371 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
1914, 16, 18sylancr 589 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
2013, 19eqtrd 2854 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
2120breq1d 5067 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
227ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝐹:𝑆⟶ℂ)
2322, 11ffvelrnd 6845 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (𝐹‘1) ∈ ℂ)
247adantr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
2524ffvelrnda 6844 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (𝐹𝑦) ∈ ℂ)
2617cnmetdval 23371 . . . . . . . . . . . . . . 15 (((𝐹‘1) ∈ ℂ ∧ (𝐹𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) = (abs‘((𝐹‘1) − (𝐹𝑦))))
2723, 25, 26syl2anc 586 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) = (abs‘((𝐹‘1) − (𝐹𝑦))))
2827breq1d 5067 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟))
2921, 28imbi12d 347 . . . . . . . . . . . 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 3194 . . . . . . . . . . 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 3295 . . . . . . . . . 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 259 . . . . . . . . 9 ((𝜑𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))
3332ralrimiva 3180 . . . . . . . 8 (𝜑 → ∀𝑟 ∈ ℝ+𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))
34 cnxmet 23373 . . . . . . . . . 10 (abs ∘ − ) ∈ (∞Met‘ℂ)
35 xmetres2 22963 . . . . . . . . . 10 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
3634, 15, 35mp2an 690 . . . . . . . . 9 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
37 eqid 2819 . . . . . . . . . . . 12 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
38 eqid 2819 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
3938cnfldtopn 23382 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
40 eqid 2819 . . . . . . . . . . . 12 (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
4137, 39, 40metrest 23126 . . . . . . . . . . 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 23143 . . . . . . . . 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 1458 . . . . . . . 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 487 . . . . . . 7 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
4847fveq2d 6667 . . . . . 6 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
4946, 48eleqtrrd 2914 . . . . 5 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
50 eldifsn 4711 . . . . . . 7 (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦𝑆𝑦 ≠ 1))
519simprd 498 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
52 abscl 14630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
5352adantl 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ)
5453a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) ∈ ℝ))
55 absge0 14639 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → 0 ≤ (abs‘𝑤))
5655adantl 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → 0 ≤ (abs‘𝑤))
5756a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → 0 ≤ (abs‘𝑤)))
581, 2abelthlem1 25011 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
5958adantr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
6053rexrd 10683 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ*)
61 1re 10633 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℝ
62 rexr 10679 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ → 1 ∈ ℝ*)
6361, 62mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → 1 ∈ ℝ*)
64 iccssxr 12811 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0[,]+∞) ⊆ ℝ*
65 eqid 2819 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))
66 eqid 2819 . . . . . . . . . . . . . . . . . . . . . . . . . 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 24997 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
6864, 67sseldi 3963 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
6968adantr 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
70 xrltletr 12542 . . . . . . . . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . . . . . . . . . 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 1123 . . . . . . . . . . . . . . . . . . . 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 10625 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℂ
7517cnmetdval 23371 . . . . . . . . . . . . . . . . . . . . . . . 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 14678 . . . . . . . . . . . . . . . . . . . . . . . 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 10898 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
8079fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
8176, 78, 803eqtrd 2858 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
8281breq1d 5067 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
8382adantl 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
84 0re 10635 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ
85 elico2 12792 . . . . . . . . . . . . . . . . . . . . 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 589 . . . . . . . . . . . . . . . . . . . 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 296 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 → (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
8887imdistanda 574 . . . . . . . . . . . . . . . . . 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 10692 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ*
90 elbl 22990 . . . . . . . . . . . . . . . . . . 19 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1)))
9134, 74, 89, 90mp3an 1454 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
92 absf 14689 . . . . . . . . . . . . . . . . . . 19 abs:ℂ⟶ℝ
93 ffn 6507 . . . . . . . . . . . . . . . . . . 19 (abs:ℂ⟶ℝ → abs Fn ℂ)
94 elpreima 6821 . . . . . . . . . . . . . . . . . . 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 298 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) → 𝑤 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
9796ssrdv 3971 . . . . . . . . . . . . . . . 16 (𝜑 → (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
9851, 97sstrd 3975 . . . . . . . . . . . . . . 15 (𝜑 → (𝑆 ∖ {1}) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
9998resmptd 5901 . . . . . . . . . . . . . 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 5842 . . . . . . . . . . . . . . 15 (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1}))
101 difss 4106 . . . . . . . . . . . . . . . 16 (𝑆 ∖ {1}) ⊆ 𝑆
102 resmpt 5898 . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . 14 (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
10599, 104syl6eqr 2872 . . . . . . . . . . . . 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 5942 . . . . . . . . . . . . . . . . . . 19 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
10792fdmi 6517 . . . . . . . . . . . . . . . . . . 19 dom abs = ℂ
108106, 107sseqtri 4001 . . . . . . . . . . . . . . . . . 18 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
109108sseli 3961 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → 𝑥 ∈ ℂ)
11065pserval2 24991 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗) = ((𝐴𝑗) · (𝑥𝑗)))
111110sumeq2dv 15052 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℂ → Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗) = Σ𝑗 ∈ ℕ0 ((𝐴𝑗) · (𝑥𝑗)))
112 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
113 oveq2 7156 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝑥𝑛) = (𝑥𝑗))
114112, 113oveq12d 7166 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((𝐴𝑛) · (𝑥𝑛)) = ((𝐴𝑗) · (𝑥𝑗)))
115114cbvsumv 15045 . . . . . . . . . . . . . . . . . 18 Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)) = Σ𝑗 ∈ ℕ0 ((𝐴𝑗) · (𝑥𝑗))
116111, 115syl6reqr 2873 . . . . . . . . . . . . . . . . 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 5148 . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . 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 25006 . . . . . . . . . . . . . 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 23497 . . . . . . . . . . . . . 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 2912 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
125124adantr 483 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
126101, 15sstri 3974 . . . . . . . . . . . 12 (𝑆 ∖ {1}) ⊆ ℂ
127 ssid 3987 . . . . . . . . . . . 12 ℂ ⊆ ℂ
128 eqid 2819 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
12938cnfldtopon 23383 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
130129toponrestid 21521 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
13138, 128, 130cncfcn 23509 . . . . . . . . . . . 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 2921 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
134 simpr 487 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
135 resttopon 21761 . . . . . . . . . . . . 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 21516 . . . . . . . . . . 11 (𝑆 ∖ {1}) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
138137cncnpi 21878 . . . . . . . . . 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 586 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
14038cnfldtop 23384 . . . . . . . . . . . 12 (TopOpen‘ℂfld) ∈ Top
141 cnex 10610 . . . . . . . . . . . . 13 ℂ ∈ V
142141, 15ssexi 5217 . . . . . . . . . . . 12 𝑆 ∈ V
143 restabs 21765 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})))
144140, 101, 142, 143mp3an 1454 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
145144oveq1i 7158 . . . . . . . . . 10 ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))
146145fveq1i 6664 . . . . . . . . 9 (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)
147139, 146eleqtrrdi 2922 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
148 resttop 21760 . . . . . . . . . . 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 4734 . . . . . . . . . . . . 13 (𝜑 → {1} ⊆ 𝑆)
15338cnfldhaus 23385 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ Haus
154 unicntop 23386 . . . . . . . . . . . . . . . 16 ℂ = (TopOpen‘ℂfld)
155154sncld 21971 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂfld)))
156153, 14, 155mp2an 690 . . . . . . . . . . . . . 14 {1} ∈ (Clsd‘(TopOpen‘ℂfld))
157154restcldi 21773 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
15815, 156, 157mp3an12 1444 . . . . . . . . . . . . 13 ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
159154restuni 21762 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
160140, 15, 159mp2an 690 . . . . . . . . . . . . . 14 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
161160cldopn 21631 . . . . . . . . . . . . 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 21666 . . . . . . . . . . . . 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 220 . . . . . . . . . . 11 (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
166165eleq2d 2896 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
167166biimpar 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})))
1687adantr 483 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
169160, 154cnprest 21889 . . . . . . . . 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 836 . . . . . . . 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 259 . . . . . . 7 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
17250, 171sylan2br 596 . . . . . 6 ((𝜑 ∧ (𝑦𝑆𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
173172anassrs 470 . . . . 5 (((𝜑𝑦𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
17449, 173pm2.61dane 3102 . . . 4 ((𝜑𝑦𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
175174ralrimiva 3180 . . 3 (𝜑 → ∀𝑦𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
176 resttopon 21761 . . . . 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 21880 . . . 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 585 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
181 eqid 2819 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
18238, 181, 130cncfcn 23509 . . 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 2922 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  {crab 3140  Vcvv 3493   ∖ cdif 3931   ⊆ wss 3934  ifcif 4465  {csn 4559  ∪ cuni 4830   class class class wbr 5057   ↦ cmpt 5137   × cxp 5546  ◡ccnv 5547  dom cdm 5548   ↾ cres 5550   “ cima 5551   ∘ ccom 5552   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  supcsup 8896  ℂcc 10527  ℝcr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664  ℝ*cxr 10666   < clt 10667   ≤ cle 10668   − cmin 10862   / cdiv 11289  2c2 11684  ℕ0cn0 11889  ℝ+crp 12381  [,)cico 12732  [,]cicc 12733  seqcseq 13361  ↑cexp 13421  abscabs 14585   ⇝ cli 14833  Σcsu 15034   ↾t crest 16686  TopOpenctopn 16687  ∞Metcxmet 20522  ballcbl 20524  MetOpencmopn 20527  ℂfldccnfld 20537  Topctop 21493  TopOnctopon 21510  Clsdccld 21616  intcnt 21617   Cn ccn 21824   CnP ccnp 21825  Hauscha 21908  –cn→ccncf 23476 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-mulg 18217  df-cntz 18439  df-cmn 18900  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-cnfld 20538  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-ntr 21620  df-cn 21827  df-cnp 21828  df-t1 21914  df-haus 21915  df-tx 22162  df-hmeo 22355  df-xms 22922  df-ms 22923  df-tms 22924  df-cncf 23478  df-ulm 24957 This theorem is referenced by:  abelth2  25022
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