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Theorem psercn 24628
Description: An infinite series converges to a continuous function on the open disk of radius 𝑅, where 𝑅 is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-Mar-2015.)
Hypotheses
Ref Expression
pserf.g 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
pserf.f 𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
pserf.a (𝜑𝐴:ℕ0⟶ℂ)
pserf.r 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )
psercn.s 𝑆 = (abs “ (0[,)𝑅))
psercn.m 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))
Assertion
Ref Expression
psercn (𝜑𝐹 ∈ (𝑆cn→ℂ))
Distinct variable groups:   𝑗,𝑎,𝑛,𝑟,𝑥,𝑦,𝐴   𝑗,𝑀,𝑦   𝑗,𝐺,𝑟,𝑦   𝑆,𝑎,𝑗,𝑦   𝐹,𝑎   𝜑,𝑎,𝑗,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑛,𝑟)   𝑅(𝑥,𝑦,𝑗,𝑛,𝑟,𝑎)   𝑆(𝑥,𝑛,𝑟)   𝐹(𝑥,𝑦,𝑗,𝑛,𝑟)   𝐺(𝑥,𝑛,𝑎)   𝑀(𝑥,𝑛,𝑟,𝑎)

Proof of Theorem psercn
Dummy variables 𝑘 𝑠 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sumex 14835 . . . . . 6 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
21rgenw 3106 . . . . 5 𝑦𝑆 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
3 pserf.f . . . . . 6 𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
43fnmpt 6268 . . . . 5 (∀𝑦𝑆 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V → 𝐹 Fn 𝑆)
52, 4mp1i 13 . . . 4 (𝜑𝐹 Fn 𝑆)
6 psercn.s . . . . . . . . . . 11 𝑆 = (abs “ (0[,)𝑅))
7 cnvimass 5741 . . . . . . . . . . . 12 (abs “ (0[,)𝑅)) ⊆ dom abs
8 absf 14491 . . . . . . . . . . . . 13 abs:ℂ⟶ℝ
98fdmi 6303 . . . . . . . . . . . 12 dom abs = ℂ
107, 9sseqtri 3856 . . . . . . . . . . 11 (abs “ (0[,)𝑅)) ⊆ ℂ
116, 10eqsstri 3854 . . . . . . . . . 10 𝑆 ⊆ ℂ
1211a1i 11 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
1312sselda 3821 . . . . . . . 8 ((𝜑𝑎𝑆) → 𝑎 ∈ ℂ)
14 0cn 10370 . . . . . . . . . . 11 0 ∈ ℂ
15 eqid 2778 . . . . . . . . . . . 12 (abs ∘ − ) = (abs ∘ − )
1615cnmetdval 22993 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
1714, 13, 16sylancr 581 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18 abssub 14480 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
1914, 13, 18sylancr 581 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
2013subid1d 10725 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑎 − 0) = 𝑎)
2120fveq2d 6452 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
2217, 19, 213eqtrd 2818 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23 breq2 4892 . . . . . . . . . . 11 ((((abs‘𝑎) + 𝑅) / 2) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2) ↔ (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))))
24 breq2 4892 . . . . . . . . . . 11 (((abs‘𝑎) + 1) = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)) → ((abs‘𝑎) < ((abs‘𝑎) + 1) ↔ (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))))
25 simpr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝑆) → 𝑎𝑆)
2625, 6syl6eleq 2869 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑎 ∈ (abs “ (0[,)𝑅)))
27 ffn 6293 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
28 elpreima 6602 . . . . . . . . . . . . . . . . . 18 (abs Fn ℂ → (𝑎 ∈ (abs “ (0[,)𝑅)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅))))
298, 27, 28mp2b 10 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (abs “ (0[,)𝑅)) ↔ (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
3026, 29sylib 210 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
3130simprd 491 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32 0re 10380 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
33 iccssxr 12573 . . . . . . . . . . . . . . . . 17 (0[,]+∞) ⊆ ℝ*
34 pserf.g . . . . . . . . . . . . . . . . . . 19 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
35 pserf.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴:ℕ0⟶ℂ)
36 pserf.r . . . . . . . . . . . . . . . . . . 19 𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺𝑟)) ∈ dom ⇝ }, ℝ*, < )
3734, 35, 36radcnvcl 24619 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ (0[,]+∞))
3837adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑅 ∈ (0[,]+∞))
3933, 38sseldi 3819 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → 𝑅 ∈ ℝ*)
40 elico2 12554 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
4132, 39, 40sylancr 581 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
4231, 41mpbid 224 . . . . . . . . . . . . . 14 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
4342simp3d 1135 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑅)
4443adantr 474 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
4513abscld 14590 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ ℝ)
46 avglt1 11625 . . . . . . . . . . . . 13 (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
4745, 46sylan 575 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
4844, 47mpbid 224 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
4945ltp1d 11311 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5049adantr 474 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5123, 24, 48, 50ifbothda 4344 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘𝑎) < if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1)))
52 psercn.m . . . . . . . . . 10 𝑀 = if(𝑅 ∈ ℝ, (((abs‘𝑎) + 𝑅) / 2), ((abs‘𝑎) + 1))
5351, 52syl6breqr 4930 . . . . . . . . 9 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑀)
5422, 53eqbrtrd 4910 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55 cnxmet 22995 . . . . . . . . . 10 (abs ∘ − ) ∈ (∞Met‘ℂ)
5655a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → (abs ∘ − ) ∈ (∞Met‘ℂ))
5714a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → 0 ∈ ℂ)
5834, 3, 35, 36, 6, 52psercnlem1 24627 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))
5958simp1d 1133 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ+)
6059rpxrd 12187 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ*)
61 elbl 22612 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6256, 57, 60, 61syl3anc 1439 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6313, 54, 62mpbir2and 703 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
64 fvres 6467 . . . . . . 7 (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹𝑎))
6563, 64syl 17 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹𝑎))
663reseq1i 5640 . . . . . . . . . 10 (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀))
6734, 3, 35, 36, 6, 58psercnlem2 24626 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)) ∧ (abs “ (0[,]𝑀)) ⊆ 𝑆))
6867simp2d 1134 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)))
6967simp3d 1135 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs “ (0[,]𝑀)) ⊆ 𝑆)
7068, 69sstrd 3831 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
7170resmptd 5704 . . . . . . . . . 10 ((𝜑𝑎𝑆) → ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
7266, 71syl5eq 2826 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
73 eqid 2778 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
7435adantr 474 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝐴:ℕ0⟶ℂ)
75 fveq2 6448 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐺𝑘) = (𝐺𝑦))
7675seqeq3d 13132 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 → seq0( + , (𝐺𝑘)) = seq0( + , (𝐺𝑦)))
7776fveq1d 6450 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → (seq0( + , (𝐺𝑘))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑠))
7877cbvmptv 4987 . . . . . . . . . . . 12 (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠))
79 fveq2 6448 . . . . . . . . . . . . 13 (𝑠 = 𝑖 → (seq0( + , (𝐺𝑦))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑖))
8079mpteq2dv 4982 . . . . . . . . . . . 12 (𝑠 = 𝑖 → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8178, 80syl5eq 2826 . . . . . . . . . . 11 (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8281cbvmptv 4987 . . . . . . . . . 10 (𝑠 ∈ ℕ0 ↦ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠))) = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8359rpred 12186 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ)
8458simp3d 1135 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 < 𝑅)
8534, 73, 74, 36, 82, 83, 84, 68psercn2 24625 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
8672, 85eqeltrd 2859 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
87 cncff 23115 . . . . . . . 8 ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
8886, 87syl 17 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
8988, 63ffvelrnd 6626 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
9065, 89eqeltrrd 2860 . . . . 5 ((𝜑𝑎𝑆) → (𝐹𝑎) ∈ ℂ)
9190ralrimiva 3148 . . . 4 (𝜑 → ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ)
92 ffnfv 6654 . . . 4 (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ))
935, 91, 92sylanbrc 578 . . 3 (𝜑𝐹:𝑆⟶ℂ)
9470, 11syl6ss 3833 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
95 ssid 3842 . . . . . . . . 9 ℂ ⊆ ℂ
96 eqid 2778 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
97 eqid 2778 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
9896cnfldtopon 23005 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9998toponrestid 21144 . . . . . . . . . 10 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
10096, 97, 99cncfcn 23131 . . . . . . . . 9 (((0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10194, 95, 100sylancl 580 . . . . . . . 8 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10286, 101eleqtrd 2861 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10396cnfldtop 23006 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ Top
10498toponunii 21139 . . . . . . . . . 10 ℂ = (TopOpen‘ℂfld)
105104restuni 21385 . . . . . . . . 9 (((TopOpen‘ℂfld) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ) → (0(ball‘(abs ∘ − ))𝑀) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
106103, 94, 105sylancr 581 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
10763, 106eleqtrd 2861 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
108 eqid 2778 . . . . . . . 8 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
109108cncnpi 21501 . . . . . . 7 (((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)) ∧ 𝑎 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
110102, 107, 109syl2anc 579 . . . . . 6 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
111103a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → (TopOpen‘ℂfld) ∈ Top)
112 cnex 10355 . . . . . . . . . . 11 ℂ ∈ V
113112, 11ssexi 5042 . . . . . . . . . 10 𝑆 ∈ V
114113a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑆 ∈ V)
115 restabs 21388 . . . . . . . . 9 (((TopOpen‘ℂfld) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆𝑆 ∈ V) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
116111, 70, 114, 115syl3anc 1439 . . . . . . . 8 ((𝜑𝑎𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
117116oveq1d 6939 . . . . . . 7 ((𝜑𝑎𝑆) → ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld)))
118117fveq1d 6450 . . . . . 6 ((𝜑𝑎𝑆) → (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎) = ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
119110, 118eleqtrrd 2862 . . . . 5 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
120 resttop 21383 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
121103, 113, 120mp2an 682 . . . . . . 7 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
122121a1i 11 . . . . . 6 ((𝜑𝑎𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
123 df-ss 3806 . . . . . . . . . 10 ((0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ↔ ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12470, 123sylib 210 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12596cnfldtopn 23004 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
126125blopn 22724 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
12756, 57, 60, 126syl3anc 1439 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
128 elrestr 16486 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V ∧ (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld)) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
129111, 114, 127, 128syl3anc 1439 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
130124, 129eqeltrrd 2860 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
131 isopn3i 21305 . . . . . . . 8 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
132121, 130, 131sylancr 581 . . . . . . 7 ((𝜑𝑎𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
13363, 132eleqtrrd 2862 . . . . . 6 ((𝜑𝑎𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
13493adantr 474 . . . . . 6 ((𝜑𝑎𝑆) → 𝐹:𝑆⟶ℂ)
135104restuni 21385 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
136103, 11, 135mp2an 682 . . . . . . 7 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
137136, 104cnprest 21512 . . . . . 6 (((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆) ∧ (𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) ∧ 𝐹:𝑆⟶ℂ)) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎) ↔ (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎)))
138122, 70, 133, 134, 137syl22anc 829 . . . . 5 ((𝜑𝑎𝑆) → (𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎) ↔ (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎)))
139119, 138mpbird 249 . . . 4 ((𝜑𝑎𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
140139ralrimiva 3148 . . 3 (𝜑 → ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
141 resttopon 21384 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
14298, 11, 141mp2an 682 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
143 cncnp 21503 . . . 4 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))))
144142, 98, 143mp2an 682 . . 3 (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎)))
14593, 140, 144sylanbrc 578 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
146 eqid 2778 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
14796, 146, 99cncfcn 23131 . . 3 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
14811, 95, 147mp2an 682 . 2 (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
149145, 148syl6eleqr 2870 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  {crab 3094  Vcvv 3398  cin 3791  wss 3792  ifcif 4307   cuni 4673   class class class wbr 4888  cmpt 4967  ccnv 5356  dom cdm 5357  cres 5359  cima 5360  ccom 5361   Fn wfn 6132  wf 6133  cfv 6137  (class class class)co 6924  supcsup 8636  cc 10272  cr 10273  0cc0 10274  1c1 10275   + caddc 10277   · cmul 10279  +∞cpnf 10410  *cxr 10412   < clt 10413  cle 10414  cmin 10608   / cdiv 11035  2c2 11435  0cn0 11647  +crp 12142  [,)cico 12494  [,]cicc 12495  seqcseq 13124  cexp 13183  abscabs 14387  cli 14632  Σcsu 14833  t crest 16478  TopOpenctopn 16479  ∞Metcxmet 20138  ballcbl 20140  fldccnfld 20153  Topctop 21116  TopOnctopon 21133  intcnt 21240   Cn ccn 21447   CnP ccnp 21448  cnccncf 23098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352  ax-addf 10353  ax-mulf 10354
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-om 7346  df-1st 7447  df-2nd 7448  df-supp 7579  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-fsupp 8566  df-fi 8607  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-4 11445  df-5 11446  df-6 11447  df-7 11448  df-8 11449  df-9 11450  df-n0 11648  df-z 11734  df-dec 11851  df-uz 11998  df-q 12101  df-rp 12143  df-xneg 12262  df-xadd 12263  df-xmul 12264  df-ico 12498  df-icc 12499  df-fz 12649  df-fzo 12790  df-fl 12917  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-limsup 14619  df-clim 14636  df-rlim 14637  df-sum 14834  df-struct 16268  df-ndx 16269  df-slot 16270  df-base 16272  df-sets 16273  df-ress 16274  df-plusg 16362  df-mulr 16363  df-starv 16364  df-sca 16365  df-vsca 16366  df-ip 16367  df-tset 16368  df-ple 16369  df-ds 16371  df-unif 16372  df-hom 16373  df-cco 16374  df-rest 16480  df-topn 16481  df-0g 16499  df-gsum 16500  df-topgen 16501  df-pt 16502  df-prds 16505  df-xrs 16559  df-qtop 16564  df-imas 16565  df-xps 16567  df-mre 16643  df-mrc 16644  df-acs 16646  df-mgm 17639  df-sgrp 17681  df-mnd 17692  df-submnd 17733  df-mulg 17939  df-cntz 18144  df-cmn 18592  df-psmet 20145  df-xmet 20146  df-met 20147  df-bl 20148  df-mopn 20149  df-cnfld 20154  df-top 21117  df-topon 21134  df-topsp 21156  df-bases 21169  df-ntr 21243  df-cn 21450  df-cnp 21451  df-tx 21785  df-hmeo 21978  df-xms 22544  df-ms 22545  df-tms 22546  df-cncf 23100  df-ulm 24579
This theorem is referenced by:  pserdvlem2  24630  pserdv  24631  abelth  24643  logtayl  24854
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