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Theorem psercn 26471
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 15725 . . . . . 6 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
21rgenw 3064 . . . . 5 𝑦𝑆 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
3 pserf.f . . . . . 6 𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
43fnmpt 6707 . . . . 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 6099 . . . . . . . . . . . 12 (abs “ (0[,)𝑅)) ⊆ dom abs
8 absf 15377 . . . . . . . . . . . . 13 abs:ℂ⟶ℝ
98fdmi 6746 . . . . . . . . . . . 12 dom abs = ℂ
107, 9sseqtri 4031 . . . . . . . . . . 11 (abs “ (0[,)𝑅)) ⊆ ℂ
116, 10eqsstri 4029 . . . . . . . . . 10 𝑆 ⊆ ℂ
1211a1i 11 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
1312sselda 3982 . . . . . . . 8 ((𝜑𝑎𝑆) → 𝑎 ∈ ℂ)
14 0cn 11254 . . . . . . . . . . 11 0 ∈ ℂ
15 eqid 2736 . . . . . . . . . . . 12 (abs ∘ − ) = (abs ∘ − )
1615cnmetdval 24792 . . . . . . . . . . 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 15366 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
1914, 13, 18sylancr 587 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
2013subid1d 11610 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑎 − 0) = 𝑎)
2120fveq2d 6909 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
2217, 19, 213eqtrd 2780 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23 breq2 5146 . . . . . . . . . . 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 5146 . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝑆) → 𝑎𝑆)
2625, 6eleqtrdi 2850 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑎 ∈ (abs “ (0[,)𝑅)))
27 ffn 6735 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
28 elpreima 7077 . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
3130simprd 495 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32 0re 11264 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
33 iccssxr 13471 . . . . . . . . . . . . . . . . 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 26461 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ (0[,]+∞))
3837adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑅 ∈ (0[,]+∞))
3933, 38sselid 3980 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → 𝑅 ∈ ℝ*)
40 elico2 13452 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 14 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
4342simp3d 1144 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑅)
4443adantr 480 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
4513abscld 15476 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ ℝ)
46 avglt1 12506 . . . . . . . . . . . . 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 232 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
4945ltp1d 12199 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5049adantr 480 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5123, 24, 48, 50ifbothda 4563 . . . . . . . . . 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 5184 . . . . . . . . 9 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑀)
5422, 53eqbrtrd 5164 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55 cnxmet 24794 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
5634, 3, 35, 36, 6, 52psercnlem1 26470 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))
5756simp1d 1142 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ+)
5857rpxrd 13079 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ*)
59 elbl 24399 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6055, 14, 58, 59mp3an12i 1466 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6113, 54, 60mpbir2and 713 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀))
6261fvresd 6925 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹𝑎))
633reseq1i 5992 . . . . . . . . . 10 (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀))
6434, 3, 35, 36, 6, 56psercnlem2 26469 . . . . . . . . . . . . 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 3993 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
6867resmptd 6057 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝐴:ℕ0⟶ℂ)
72 fveq2 6905 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐺𝑘) = (𝐺𝑦))
7372seqeq3d 14051 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 → seq0( + , (𝐺𝑘)) = seq0( + , (𝐺𝑦)))
7473fveq1d 6907 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → (seq0( + , (𝐺𝑘))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑠))
7574cbvmptv 5254 . . . . . . . . . . . 12 (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠))
76 fveq2 6905 . . . . . . . . . . . . 13 (𝑠 = 𝑖 → (seq0( + , (𝐺𝑦))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑖))
7776mpteq2dv 5243 . . . . . . . . . . . 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 5254 . . . . . . . . . 10 (𝑠 ∈ ℕ0 ↦ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠))) = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8057rpred 13078 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ)
8156simp3d 1144 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 < 𝑅)
8234, 70, 71, 36, 79, 80, 81, 65psercn2 26467 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
8369, 82eqeltrd 2840 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84 cncff 24920 . . . . . . . 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 7104 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
8762, 86eqeltrrd 2841 . . . . 5 ((𝜑𝑎𝑆) → (𝐹𝑎) ∈ ℂ)
8887ralrimiva 3145 . . . 4 (𝜑 → ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ)
89 ffnfv 7138 . . . 4 (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ))
905, 88, 89sylanbrc 583 . . 3 (𝜑𝐹:𝑆⟶ℂ)
9167, 11sstrdi 3995 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92 ssid 4005 . . . . . . . . 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 24804 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9695toponrestid 22928 . . . . . . . . . 10 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
9793, 94, 96cncfcn 24937 . . . . . . . . 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 2842 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10093cnfldtop 24805 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ Top
101 unicntop 24807 . . . . . . . . . 10 ℂ = (TopOpen‘ℂfld)
102101restuni 23171 . . . . . . . . 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 2842 . . . . . . 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 23287 . . . . . . 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 11237 . . . . . . . . . . 11 ℂ ∈ V
109108, 11ssexi 5321 . . . . . . . . . 10 𝑆 ∈ V
110109a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑆 ∈ V)
111 restabs 23174 . . . . . . . . 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 1467 . . . . . . . 8 ((𝜑𝑎𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113112oveq1d 7447 . . . . . . 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 6907 . . . . . 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 2843 . . . . 5 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116 resttop 23169 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ V) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
117100, 109, 116mp2an 692 . . . . . . 7 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top
118117a1i 11 . . . . . 6 ((𝜑𝑎𝑆) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
119 dfss2 3968 . . . . . . . . . 10 ((0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ↔ ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12067, 119sylib 218 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12193cnfldtopn 24803 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122121blopn 24514 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
12355, 14, 58, 122mp3an12i 1466 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124 elrestr 17474 . . . . . . . . . 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 1466 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126120, 125eqeltrrd 2841 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127 isopn3i 23091 . . . . . . . 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 2843 . . . . . 6 ((𝜑𝑎𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
13090adantr 480 . . . . . 6 ((𝜑𝑎𝑆) → 𝐹:𝑆⟶ℂ)
131101restuni 23171 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
132100, 11, 131mp2an 692 . . . . . . 7 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
133132, 101cnprest 23298 . . . . . 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 838 . . . . 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 257 . . . 4 ((𝜑𝑎𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
136135ralrimiva 3145 . . 3 (𝜑 → ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137 resttopon 23170 . . . . 5 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
13895, 11, 137mp2an 692 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆)
139 cncnp 23289 . . . 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 692 . . 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 24937 . . 3 ((𝑆 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
14411, 92, 143mp2an 692 . 2 (𝑆cn→ℂ) = (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld))
145141, 144eleqtrrdi 2851 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  cin 3949  wss 3950  ifcif 4524   cuni 4906   class class class wbr 5142  cmpt 5224  ccnv 5683  dom cdm 5684  cres 5686  cima 5687  ccom 5688   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  supcsup 9481  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161  +∞cpnf 11293  *cxr 11295   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  2c2 12322  0cn0 12528  +crp 13035  [,)cico 13390  [,]cicc 13391  seqcseq 14043  cexp 14103  abscabs 15274  cli 15521  Σcsu 15723  t crest 17466  TopOpenctopn 17467  ∞Metcxmet 21350  ballcbl 21352  fldccnfld 21365  Topctop 22900  TopOnctopon 22917  intcnt 23026   Cn ccn 23233   CnP ccnp 23234  cnccncf 24903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-fi 9452  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-limsup 15508  df-clim 15525  df-rlim 15526  df-sum 15724  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-submnd 18798  df-mulg 19087  df-cntz 19336  df-cmn 19801  df-psmet 21357  df-xmet 21358  df-met 21359  df-bl 21360  df-mopn 21361  df-cnfld 21366  df-top 22901  df-topon 22918  df-topsp 22940  df-bases 22954  df-ntr 23029  df-cn 23236  df-cnp 23237  df-tx 23571  df-hmeo 23764  df-xms 24331  df-ms 24332  df-tms 24333  df-cncf 24905  df-ulm 26421
This theorem is referenced by:  pserdvlem2  26473  pserdv  26474  abelth  26486  logtayl  26703
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