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Theorem psercn 25785
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 15572 . . . . . 6 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
21rgenw 3068 . . . . 5 𝑦𝑆 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
3 pserf.f . . . . . 6 𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
43fnmpt 6641 . . . . 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 6033 . . . . . . . . . . . 12 (abs “ (0[,)𝑅)) ⊆ dom abs
8 absf 15222 . . . . . . . . . . . . 13 abs:ℂ⟶ℝ
98fdmi 6680 . . . . . . . . . . . 12 dom abs = ℂ
107, 9sseqtri 3980 . . . . . . . . . . 11 (abs “ (0[,)𝑅)) ⊆ ℂ
116, 10eqsstri 3978 . . . . . . . . . 10 𝑆 ⊆ ℂ
1211a1i 11 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
1312sselda 3944 . . . . . . . 8 ((𝜑𝑎𝑆) → 𝑎 ∈ ℂ)
14 0cn 11147 . . . . . . . . . . 11 0 ∈ ℂ
15 eqid 2736 . . . . . . . . . . . 12 (abs ∘ − ) = (abs ∘ − )
1615cnmetdval 24134 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
1714, 13, 16sylancr 587 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18 abssub 15211 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
1914, 13, 18sylancr 587 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
2013subid1d 11501 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑎 − 0) = 𝑎)
2120fveq2d 6846 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
2217, 19, 213eqtrd 2780 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23 breq2 5109 . . . . . . . . . . 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 5109 . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝑆) → 𝑎𝑆)
2625, 6eleqtrdi 2848 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑎 ∈ (abs “ (0[,)𝑅)))
27 ffn 6668 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
28 elpreima 7008 . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
3130simprd 496 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32 0re 11157 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
33 iccssxr 13347 . . . . . . . . . . . . . . . . 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 25776 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ (0[,]+∞))
3837adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑅 ∈ (0[,]+∞))
3933, 38sselid 3942 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → 𝑅 ∈ ℝ*)
40 elico2 13328 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
4132, 39, 40sylancr 587 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
4231, 41mpbid 231 . . . . . . . . . . . . . 14 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
4342simp3d 1144 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑅)
4443adantr 481 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
4513abscld 15321 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ ℝ)
46 avglt1 12391 . . . . . . . . . . . . 13 (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
4745, 46sylan 580 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
4844, 47mpbid 231 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
4945ltp1d 12085 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5049adantr 481 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5123, 24, 48, 50ifbothda 4524 . . . . . . . . . 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, 52breqtrrdi 5147 . . . . . . . . 9 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑀)
5422, 53eqbrtrd 5127 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55 cnxmet 24136 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
5634, 3, 35, 36, 6, 52psercnlem1 25784 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))
5756simp1d 1142 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ+)
5857rpxrd 12958 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ*)
59 elbl 23741 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6055, 14, 58, 59mp3an12i 1465 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6113, 54, 60mpbir2and 711 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
6261fvresd 6862 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹𝑎))
633reseq1i 5933 . . . . . . . . . 10 (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀))
6434, 3, 35, 36, 6, 56psercnlem2 25783 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)) ∧ (abs “ (0[,]𝑀)) ⊆ 𝑆))
6564simp2d 1143 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)))
6664simp3d 1144 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs “ (0[,]𝑀)) ⊆ 𝑆)
6765, 66sstrd 3954 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
6867resmptd 5994 . . . . . . . . . 10 ((𝜑𝑎𝑆) → ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
6963, 68eqtrid 2788 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
70 eqid 2736 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
7135adantr 481 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝐴:ℕ0⟶ℂ)
72 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐺𝑘) = (𝐺𝑦))
7372seqeq3d 13914 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 → seq0( + , (𝐺𝑘)) = seq0( + , (𝐺𝑦)))
7473fveq1d 6844 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → (seq0( + , (𝐺𝑘))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑠))
7574cbvmptv 5218 . . . . . . . . . . . 12 (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠))
76 fveq2 6842 . . . . . . . . . . . . 13 (𝑠 = 𝑖 → (seq0( + , (𝐺𝑦))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑖))
7776mpteq2dv 5207 . . . . . . . . . . . 12 (𝑠 = 𝑖 → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
7875, 77eqtrid 2788 . . . . . . . . . . 11 (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
7978cbvmptv 5218 . . . . . . . . . 10 (𝑠 ∈ ℕ0 ↦ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠))) = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8057rpred 12957 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ)
8156simp3d 1144 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 < 𝑅)
8234, 70, 71, 36, 79, 80, 81, 65psercn2 25782 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
8369, 82eqeltrd 2838 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84 cncff 24256 . . . . . . . 8 ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
8583, 84syl 17 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)):(0(ball‘(abs ∘ − ))𝑀)⟶ℂ)
8685, 61ffvelcdmd 7036 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
8762, 86eqeltrrd 2839 . . . . 5 ((𝜑𝑎𝑆) → (𝐹𝑎) ∈ ℂ)
8887ralrimiva 3143 . . . 4 (𝜑 → ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ)
89 ffnfv 7066 . . . 4 (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ))
905, 88, 89sylanbrc 583 . . 3 (𝜑𝐹:𝑆⟶ℂ)
9167, 11sstrdi 3956 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92 ssid 3966 . . . . . . . . 9 ℂ ⊆ ℂ
93 eqid 2736 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94 eqid 2736 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
9593cnfldtopon 24146 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9695toponrestid 22270 . . . . . . . . . 10 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
9793, 94, 96cncfcn 24273 . . . . . . . . 9 (((0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
9891, 92, 97sylancl 586 . . . . . . . 8 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
9983, 98eleqtrd 2840 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10093cnfldtop 24147 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ Top
101 unicntop 24149 . . . . . . . . . 10 ℂ = (TopOpen‘ℂfld)
102101restuni 22513 . . . . . . . . 9 (((TopOpen‘ℂfld) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ) → (0(ball‘(abs ∘ − ))𝑀) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
103100, 91, 102sylancr 587 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
10461, 103eleqtrd 2840 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105 eqid 2736 . . . . . . . 8 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106105cncnpi 22629 . . . . . . 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))
10799, 104, 106syl2anc 584 . . . . . 6 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108 cnex 11132 . . . . . . . . . . 11 ℂ ∈ V
109108, 11ssexi 5279 . . . . . . . . . 10 𝑆 ∈ V
110109a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑆 ∈ V)
111 restabs 22516 . . . . . . . . 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)))
112100, 67, 110, 111mp3an2i 1466 . . . . . . . 8 ((𝜑𝑎𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113112oveq1d 7372 . . . . . . 7 ((𝜑𝑎𝑆) → ((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld)))
114113fveq1d 6844 . . . . . 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))
115107, 114eleqtrrd 2841 . . . . 5 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116 resttop 22511 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117100, 109, 116mp2an 690 . . . . . . 7 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
118117a1i 11 . . . . . 6 ((𝜑𝑎𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
119 df-ss 3927 . . . . . . . . . 10 ((0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ↔ ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12067, 119sylib 217 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12193cnfldtopn 24145 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122121blopn 23856 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
12355, 14, 58, 122mp3an12i 1465 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124 elrestr 17310 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V ∧ (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld)) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
125100, 109, 123, 124mp3an12i 1465 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126120, 125eqeltrrd 2839 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127 isopn3i 22433 . . . . . . . 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))
128117, 126, 127sylancr 587 . . . . . . 7 ((𝜑𝑎𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
12961, 128eleqtrrd 2841 . . . . . 6 ((𝜑𝑎𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
13090adantr 481 . . . . . 6 ((𝜑𝑎𝑆) → 𝐹:𝑆⟶ℂ)
131101restuni 22513 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
132100, 11, 131mp2an 690 . . . . . . 7 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
133132, 101cnprest 22640 . . . . . 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)))
134118, 67, 129, 130, 133syl22anc 837 . . . . 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)))
135115, 134mpbird 256 . . . 4 ((𝜑𝑎𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
136135ralrimiva 3143 . . 3 (𝜑 → ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137 resttopon 22512 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
13895, 11, 137mp2an 690 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139 cncnp 22631 . . . 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))))
140138, 95, 139mp2an 690 . . 3 (𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)) ↔ (𝐹:𝑆⟶ℂ ∧ ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎)))
14190, 136, 140sylanbrc 583 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142 eqid 2736 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
14393, 142, 96cncfcn 24273 . . 3 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
14411, 92, 143mp2an 690 . 2 (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145141, 144eleqtrrdi 2849 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cin 3909  wss 3910  ifcif 4486   cuni 4865   class class class wbr 5105  cmpt 5188  ccnv 5632  dom cdm 5633  cres 5635  cima 5636  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  2c2 12208  0cn0 12413  +crp 12915  [,)cico 13266  [,]cicc 13267  seqcseq 13906  cexp 13967  abscabs 15119  cli 15366  Σcsu 15570  t crest 17302  TopOpenctopn 17303  ∞Metcxmet 20781  ballcbl 20783  fldccnfld 20796  Topctop 22242  TopOnctopon 22259  intcnt 22368   Cn ccn 22575   CnP ccnp 22576  cnccncf 24239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-ntr 22371  df-cn 22578  df-cnp 22579  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-ulm 25736
This theorem is referenced by:  pserdvlem2  25787  pserdv  25788  abelth  25800  logtayl  26015
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