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Theorem abelth 24240
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 24225.) (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 24233 . . 3 (𝜑𝐹:𝑆⟶ℂ)
81, 2, 3, 4, 5, 6abelthlem9 24239 . . . . . . . . . 10 ((𝜑𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟))
91, 2, 3, 4, 5abelthlem2 24231 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ 𝑆 ∧ (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1)))
109simpld 474 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ∈ 𝑆)
1110ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 1 ∈ 𝑆)
12 simpr 476 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝑦𝑆)
1311, 12ovresd 6843 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (1(abs ∘ − )𝑦))
14 ax-1cn 10032 . . . . . . . . . . . . . . . 16 1 ∈ ℂ
15 ssrab2 3720 . . . . . . . . . . . . . . . . . 18 {𝑧 ∈ ℂ ∣ (abs‘(1 − 𝑧)) ≤ (𝑀 · (1 − (abs‘𝑧)))} ⊆ ℂ
165, 15eqsstri 3668 . . . . . . . . . . . . . . . . 17 𝑆 ⊆ ℂ
1716, 12sseldi 3634 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝑦 ∈ ℂ)
18 eqid 2651 . . . . . . . . . . . . . . . . 17 (abs ∘ − ) = (abs ∘ − )
1918cnmetdval 22621 . . . . . . . . . . . . . . . 16 ((1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
2014, 17, 19sylancr 696 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1(abs ∘ − )𝑦) = (abs‘(1 − 𝑦)))
2113, 20eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) = (abs‘(1 − 𝑦)))
2221breq1d 4695 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 ↔ (abs‘(1 − 𝑦)) < 𝑤))
237ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → 𝐹:𝑆⟶ℂ)
2423, 11ffvelrnd 6400 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (𝐹‘1) ∈ ℂ)
257adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑟 ∈ ℝ+) → 𝐹:𝑆⟶ℂ)
2625ffvelrnda 6399 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (𝐹𝑦) ∈ ℂ)
2718cnmetdval 22621 . . . . . . . . . . . . . . 15 (((𝐹‘1) ∈ ℂ ∧ (𝐹𝑦) ∈ ℂ) → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) = (abs‘((𝐹‘1) − (𝐹𝑦))))
2824, 26, 27syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) = (abs‘((𝐹‘1) − (𝐹𝑦))))
2928breq1d 4695 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟 ↔ (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟))
3022, 29imbi12d 333 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑦𝑆) → (((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟) ↔ ((abs‘(1 − 𝑦)) < 𝑤 → (abs‘((𝐹‘1) − (𝐹𝑦))) < 𝑟)))
3130ralbidva 3014 . . . . . . . . . . 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)))
3231rexbidv 3081 . . . . . . . . . 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)))
338, 32mpbird 247 . . . . . . . . 9 ((𝜑𝑟 ∈ ℝ+) → ∃𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))
3433ralrimiva 2995 . . . . . . . 8 (𝜑 → ∀𝑟 ∈ ℝ+𝑤 ∈ ℝ+𝑦𝑆 ((1((abs ∘ − ) ↾ (𝑆 × 𝑆))𝑦) < 𝑤 → ((𝐹‘1)(abs ∘ − )(𝐹𝑦)) < 𝑟))
35 cnxmet 22623 . . . . . . . . . . 11 (abs ∘ − ) ∈ (∞Met‘ℂ)
36 xmetres2 22213 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
3735, 16, 36mp2an 708 . . . . . . . . . 10 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆)
3837a1i 11 . . . . . . . . 9 (𝜑 → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
3935a1i 11 . . . . . . . . 9 (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
40 eqid 2651 . . . . . . . . . . . 12 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
41 eqid 2651 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4241cnfldtopn 22632 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
43 eqid 2651 . . . . . . . . . . . 12 (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
4440, 42, 43metrest 22376 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
4535, 16, 44mp2an 708 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
4645, 42metcnp 22393 . . . . . . . . 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))))
4738, 39, 10, 46syl3anc 1366 . . . . . . . 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))))
487, 34, 47mpbir2and 977 . . . . . . 7 (𝜑𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
4948ad2antrr 762 . . . . . 6 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
50 simpr 476 . . . . . . 7 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝑦 = 1)
5150fveq2d 6233 . . . . . 6 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘1))
5249, 51eleqtrrd 2733 . . . . 5 (((𝜑𝑦𝑆) ∧ 𝑦 = 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
53 eldifsn 4350 . . . . . . 7 (𝑦 ∈ (𝑆 ∖ {1}) ↔ (𝑦𝑆𝑦 ≠ 1))
549simprd 478 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑆 ∖ {1}) ⊆ (0(ball‘(abs ∘ − ))1))
55 abscl 14062 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘𝑤) ∈ ℝ)
5655adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ)
5756a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) ∈ ℝ))
58 absge0 14071 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → 0 ≤ (abs‘𝑤))
5958adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → 0 ≤ (abs‘𝑤))
6059a1d 25 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → 0 ≤ (abs‘𝑤)))
611, 2abelthlem1 24230 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
6261adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ ℂ) → 1 ≤ sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))
6356rexrd 10127 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → (abs‘𝑤) ∈ ℝ*)
64 1re 10077 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℝ
65 rexr 10123 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℝ → 1 ∈ ℝ*)
6664, 65mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → 1 ∈ ℝ*)
67 iccssxr 12294 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0[,]+∞) ⊆ ℝ*
68 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛)))) = (𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))
69 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . 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*, < )
7068, 1, 69radcnvcl 24216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ (0[,]+∞))
7167, 70sseldi 3634 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
7271adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ ℂ) → sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ) ∈ ℝ*)
73 xrltletr 12026 . . . . . . . . . . . . . . . . . . . . . . 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*, < )))
7463, 66, 72, 73syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . 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*, < )))
7562, 74mpan2d 710 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ ℂ) → ((abs‘𝑤) < 1 → (abs‘𝑤) < sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))
7657, 60, 753jcad 1262 . . . . . . . . . . . . . . . . . . . 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*, < ))))
77 0cn 10070 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ ℂ
7818cnmetdval 22621 . . . . . . . . . . . . . . . . . . . . . . . 24 ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
7977, 78mpan 706 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘(0 − 𝑤)))
80 abssub 14110 . . . . . . . . . . . . . . . . . . . . . . . 24 ((0 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
8177, 80mpan 706 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘(0 − 𝑤)) = (abs‘(𝑤 − 0)))
82 subid1 10339 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 ∈ ℂ → (𝑤 − 0) = 𝑤)
8382fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ ℂ → (abs‘(𝑤 − 0)) = (abs‘𝑤))
8479, 81, 833eqtrd 2689 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ∈ ℂ → (0(abs ∘ − )𝑤) = (abs‘𝑤))
8584breq1d 4695 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 ∈ ℂ → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
8685adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 ↔ (abs‘𝑤) < 1))
87 0re 10078 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℝ
88 elico2 12275 . . . . . . . . . . . . . . . . . . . . 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*, < ))))
8987, 72, 88sylancr 696 . . . . . . . . . . . . . . . . . . . 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*, < ))))
9076, 86, 893imtr4d 283 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ℂ) → ((0(abs ∘ − )𝑤) < 1 → (abs‘𝑤) ∈ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
9190imdistanda 729 . . . . . . . . . . . . . . . . . 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*, < )))))
9264rexri 10135 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ*
93 elbl 22240 . . . . . . . . . . . . . . . . . . 19 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 1 ∈ ℝ*) → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1)))
9435, 77, 92, 93mp3an 1464 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ (0(ball‘(abs ∘ − ))1) ↔ (𝑤 ∈ ℂ ∧ (0(abs ∘ − )𝑤) < 1))
95 absf 14121 . . . . . . . . . . . . . . . . . . 19 abs:ℂ⟶ℝ
96 ffn 6083 . . . . . . . . . . . . . . . . . . 19 (abs:ℂ⟶ℝ → abs Fn ℂ)
97 elpreima 6377 . . . . . . . . . . . . . . . . . . 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*, < )))))
9895, 96, 97mp2b 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*, < ))))
9991, 94, 983imtr4g 285 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑤 ∈ (0(ball‘(abs ∘ − ))1) → 𝑤 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < )))))
10099ssrdv 3642 . . . . . . . . . . . . . . . 16 (𝜑 → (0(ball‘(abs ∘ − ))1) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
10154, 100sstrd 3646 . . . . . . . . . . . . . . 15 (𝜑 → (𝑆 ∖ {1}) ⊆ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))))
102101resmptd 5487 . . . . . . . . . . . . . 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))))
1036reseq1i 5424 . . . . . . . . . . . . . . 15 (𝐹 ↾ (𝑆 ∖ {1})) = ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1}))
104 difss 3770 . . . . . . . . . . . . . . . 16 (𝑆 ∖ {1}) ⊆ 𝑆
105 resmpt 5484 . . . . . . . . . . . . . . . 16 ((𝑆 ∖ {1}) ⊆ 𝑆 → ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))))
106104, 105ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑥𝑆 ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛))) ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
107103, 106eqtri 2673 . . . . . . . . . . . . . 14 (𝐹 ↾ (𝑆 ∖ {1})) = (𝑥 ∈ (𝑆 ∖ {1}) ↦ Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)))
108102, 107syl6eqr 2703 . . . . . . . . . . . . 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})))
109 cnvimass 5520 . . . . . . . . . . . . . . . . . . 19 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ dom abs
11095fdmi 6090 . . . . . . . . . . . . . . . . . . 19 dom abs = ℂ
111109, 110sseqtri 3670 . . . . . . . . . . . . . . . . . 18 (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) ⊆ ℂ
112111sseli 3632 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (abs “ (0[,)sup({𝑟 ∈ ℝ ∣ seq0( + , ((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑟)) ∈ dom ⇝ }, ℝ*, < ))) → 𝑥 ∈ ℂ)
11368pserval2 24210 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗) = ((𝐴𝑗) · (𝑥𝑗)))
114113sumeq2dv 14477 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℂ → Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗) = Σ𝑗 ∈ ℕ0 ((𝐴𝑗) · (𝑥𝑗)))
115 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝐴𝑛) = (𝐴𝑗))
116 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑗 → (𝑥𝑛) = (𝑥𝑗))
117115, 116oveq12d 6708 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → ((𝐴𝑛) · (𝑥𝑛)) = ((𝐴𝑗) · (𝑥𝑗)))
118117cbvsumv 14470 . . . . . . . . . . . . . . . . . 18 Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)) = Σ𝑗 ∈ ℕ0 ((𝐴𝑗) · (𝑥𝑗))
119114, 118syl6reqr 2704 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℂ → Σ𝑛 ∈ ℕ0 ((𝐴𝑛) · (𝑥𝑛)) = Σ𝑗 ∈ ℕ0 (((𝑡 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑡𝑛))))‘𝑥)‘𝑗))
120112, 119syl 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))
121120mpteq2ia 4773 . . . . . . . . . . . . . . 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))
122 eqid 2651 . . . . . . . . . . . . . . 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*, < )))
123 eqid 2651 . . . . . . . . . . . . . . 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))
12468, 121, 1, 69, 122, 123psercn 24225 . . . . . . . . . . . . . 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))
125 rescncf 22747 . . . . . . . . . . . . . 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)))
126101, 124, 125sylc 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))
127108, 126eqeltrrd 2731 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
128127adantr 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((𝑆 ∖ {1})–cn→ℂ))
129104, 16sstri 3645 . . . . . . . . . . . 12 (𝑆 ∖ {1}) ⊆ ℂ
130 ssid 3657 . . . . . . . . . . . 12 ℂ ⊆ ℂ
131 eqid 2651 . . . . . . . . . . . . 13 ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
13241cnfldtop 22634 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ Top
13341cnfldtopon 22633 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
134133toponunii 20769 . . . . . . . . . . . . . . . 16 ℂ = (TopOpen‘ℂfld)
135134restid 16141 . . . . . . . . . . . . . . 15 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
136132, 135ax-mp 5 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
137136eqcomi 2660 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
13841, 131, 137cncfcn 22759 . . . . . . . . . . . 12 (((𝑆 ∖ {1}) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
139129, 130, 138mp2an 708 . . . . . . . . . . 11 ((𝑆 ∖ {1})–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld))
140128, 139syl6eleq 2740 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) Cn (TopOpen‘ℂfld)))
141 simpr 476 . . . . . . . . . 10 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ (𝑆 ∖ {1}))
142 resttopon 21013 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝑆 ∖ {1}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1})))
143133, 129, 142mp2an 708 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) ∈ (TopOn‘(𝑆 ∖ {1}))
144143toponunii 20769 . . . . . . . . . . 11 (𝑆 ∖ {1}) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
145144cncnpi 21130 . . . . . . . . . 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))
146140, 141, 145syl2anc 694 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
147 cnex 10055 . . . . . . . . . . . . 13 ℂ ∈ V
148147, 16ssexi 4836 . . . . . . . . . . . 12 𝑆 ∈ V
149 restabs 21017 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})))
150132, 104, 148, 149mp3an 1464 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) = ((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1}))
151150oveq1i 6700 . . . . . . . . . 10 ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))
152151fveq1i 6230 . . . . . . . . 9 (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦) = ((((TopOpen‘ℂfld) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦)
153146, 152syl6eleqr 2741 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝐹 ↾ (𝑆 ∖ {1})) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (𝑆 ∖ {1})) CnP (TopOpen‘ℂfld))‘𝑦))
154 resttop 21012 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
155132, 148, 154mp2an 708 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
156155a1i 11 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
157104a1i 11 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → (𝑆 ∖ {1}) ⊆ 𝑆)
15810snssd 4372 . . . . . . . . . . . . 13 (𝜑 → {1} ⊆ 𝑆)
15941cnfldhaus 22635 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ Haus
160134sncld 21223 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Haus ∧ 1 ∈ ℂ) → {1} ∈ (Clsd‘(TopOpen‘ℂfld)))
161159, 14, 160mp2an 708 . . . . . . . . . . . . . 14 {1} ∈ (Clsd‘(TopOpen‘ℂfld))
162134restcldi 21025 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ℂ ∧ {1} ∈ (Clsd‘(TopOpen‘ℂfld)) ∧ {1} ⊆ 𝑆) → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
16316, 161, 162mp3an12 1454 . . . . . . . . . . . . 13 ({1} ⊆ 𝑆 → {1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)))
164134restuni 21014 . . . . . . . . . . . . . . 15 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
165132, 16, 164mp2an 708 . . . . . . . . . . . . . 14 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
166165cldopn 20883 . . . . . . . . . . . . 13 ({1} ∈ (Clsd‘((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
167158, 163, 1663syl 18 . . . . . . . . . . . 12 (𝜑 → (𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
168165isopn3 20918 . . . . . . . . . . . . 13 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (𝑆 ∖ {1}) ⊆ 𝑆) → ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1})))
169155, 104, 168mp2an 708 . . . . . . . . . . . 12 ((𝑆 ∖ {1}) ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
170167, 169sylib 208 . . . . . . . . . . 11 (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) = (𝑆 ∖ {1}))
171170eleq2d 2716 . . . . . . . . . 10 (𝜑 → (𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})) ↔ 𝑦 ∈ (𝑆 ∖ {1})))
172171biimpar 501 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝑦 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(𝑆 ∖ {1})))
1737adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝐹:𝑆⟶ℂ)
174165, 134cnprest 21141 . . . . . . . . 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)))
175156, 157, 172, 173, 174syl22anc 1367 . . . . . . . 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)))
176153, 175mpbird 247 . . . . . . 7 ((𝜑𝑦 ∈ (𝑆 ∖ {1})) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
17753, 176sylan2br 492 . . . . . 6 ((𝜑 ∧ (𝑦𝑆𝑦 ≠ 1)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
178177anassrs 681 . . . . 5 (((𝜑𝑦𝑆) ∧ 𝑦 ≠ 1) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
17952, 178pm2.61dane 2910 . . . 4 ((𝜑𝑦𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
180179ralrimiva 2995 . . 3 (𝜑 → ∀𝑦𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦))
181 resttopon 21013 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
182133, 16, 181mp2an 708 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
183 cncnp 21132 . . . 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))))
184182, 133, 183mp2an 708 . . 3 (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑦𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑦)))
1857, 180, 184sylanbrc 699 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
186 eqid 2651 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
18741, 186, 137cncfcn 22759 . . 3 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
18816, 130, 187mp2an 708 . 2 (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
189185, 188syl6eleqr 2741 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  ifcif 4119  {csn 4210   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  2c2 11108  0cn0 11330  +crp 11870  [,)cico 12215  [,]cicc 12216  seqcseq 12841  cexp 12900  abscabs 14018  cli 14259  Σcsu 14460  t crest 16128  TopOpenctopn 16129  ∞Metcxmt 19779  ballcbl 19781  MetOpencmopn 19784  fldccnfld 19794  Topctop 20746  TopOnctopon 20763  Clsdccld 20868  intcnt 20869   Cn ccn 21076   CnP ccnp 21077  Hauscha 21160  cnccncf 22726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cn 21079  df-cnp 21080  df-t1 21166  df-haus 21167  df-tx 21413  df-hmeo 21606  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-ulm 24176
This theorem is referenced by:  abelth2  24241
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