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Theorem psercn 26485
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 15721 . . . . . 6 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
21rgenw 3063 . . . . 5 𝑦𝑆 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
3 pserf.f . . . . . 6 𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
43fnmpt 6709 . . . . 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 6102 . . . . . . . . . . . 12 (abs “ (0[,)𝑅)) ⊆ dom abs
8 absf 15373 . . . . . . . . . . . . 13 abs:ℂ⟶ℝ
98fdmi 6748 . . . . . . . . . . . 12 dom abs = ℂ
107, 9sseqtri 4032 . . . . . . . . . . 11 (abs “ (0[,)𝑅)) ⊆ ℂ
116, 10eqsstri 4030 . . . . . . . . . 10 𝑆 ⊆ ℂ
1211a1i 11 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
1312sselda 3995 . . . . . . . 8 ((𝜑𝑎𝑆) → 𝑎 ∈ ℂ)
14 0cn 11251 . . . . . . . . . . 11 0 ∈ ℂ
15 eqid 2735 . . . . . . . . . . . 12 (abs ∘ − ) = (abs ∘ − )
1615cnmetdval 24807 . . . . . . . . . . 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 15362 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
1914, 13, 18sylancr 587 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
2013subid1d 11607 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑎 − 0) = 𝑎)
2120fveq2d 6911 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
2217, 19, 213eqtrd 2779 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23 breq2 5152 . . . . . . . . . . 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 5152 . . . . . . . . . . 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 2849 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑎 ∈ (abs “ (0[,)𝑅)))
27 ffn 6737 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
28 elpreima 7078 . . . . . . . . . . . . . . . . . 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 11261 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
33 iccssxr 13467 . . . . . . . . . . . . . . . . 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 26475 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ (0[,]+∞))
3837adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑅 ∈ (0[,]+∞))
3933, 38sselid 3993 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → 𝑅 ∈ ℝ*)
40 elico2 13448 . . . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑅)
4443adantr 480 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
4513abscld 15472 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ ℝ)
46 avglt1 12502 . . . . . . . . . . . . 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 12196 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5049adantr 480 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5123, 24, 48, 50ifbothda 4569 . . . . . . . . . 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 5190 . . . . . . . . 9 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑀)
5422, 53eqbrtrd 5170 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55 cnxmet 24809 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
5634, 3, 35, 36, 6, 52psercnlem1 26484 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))
5756simp1d 1141 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ+)
5857rpxrd 13076 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ*)
59 elbl 24414 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6055, 14, 58, 59mp3an12i 1464 . . . . . . . 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 6927 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹𝑎))
633reseq1i 5996 . . . . . . . . . 10 (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀))
6434, 3, 35, 36, 6, 56psercnlem2 26483 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)) ∧ (abs “ (0[,]𝑀)) ⊆ 𝑆))
6564simp2d 1142 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)))
6664simp3d 1143 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs “ (0[,]𝑀)) ⊆ 𝑆)
6765, 66sstrd 4006 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
6867resmptd 6060 . . . . . . . . . 10 ((𝜑𝑎𝑆) → ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
6963, 68eqtrid 2787 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
70 eqid 2735 . . . . . . . . . 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 6907 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐺𝑘) = (𝐺𝑦))
7372seqeq3d 14047 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 → seq0( + , (𝐺𝑘)) = seq0( + , (𝐺𝑦)))
7473fveq1d 6909 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → (seq0( + , (𝐺𝑘))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑠))
7574cbvmptv 5261 . . . . . . . . . . . 12 (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠))
76 fveq2 6907 . . . . . . . . . . . . 13 (𝑠 = 𝑖 → (seq0( + , (𝐺𝑦))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑖))
7776mpteq2dv 5250 . . . . . . . . . . . 12 (𝑠 = 𝑖 → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
7875, 77eqtrid 2787 . . . . . . . . . . 11 (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
7978cbvmptv 5261 . . . . . . . . . 10 (𝑠 ∈ ℕ0 ↦ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠))) = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8057rpred 13075 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ)
8156simp3d 1143 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 < 𝑅)
8234, 70, 71, 36, 79, 80, 81, 65psercn2 26481 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
8369, 82eqeltrd 2839 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84 cncff 24933 . . . . . . . 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 7105 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
8762, 86eqeltrrd 2840 . . . . 5 ((𝜑𝑎𝑆) → (𝐹𝑎) ∈ ℂ)
8887ralrimiva 3144 . . . 4 (𝜑 → ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ)
89 ffnfv 7139 . . . 4 (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ))
905, 88, 89sylanbrc 583 . . 3 (𝜑𝐹:𝑆⟶ℂ)
9167, 11sstrdi 4008 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92 ssid 4018 . . . . . . . . 9 ℂ ⊆ ℂ
93 eqid 2735 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94 eqid 2735 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
9593cnfldtopon 24819 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9695toponrestid 22943 . . . . . . . . . 10 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
9793, 94, 96cncfcn 24950 . . . . . . . . 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 2841 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10093cnfldtop 24820 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ Top
101 unicntop 24822 . . . . . . . . . 10 ℂ = (TopOpen‘ℂfld)
102101restuni 23186 . . . . . . . . 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 2841 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105 eqid 2735 . . . . . . . 8 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106105cncnpi 23302 . . . . . . 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 11234 . . . . . . . . . . 11 ℂ ∈ V
109108, 11ssexi 5328 . . . . . . . . . 10 𝑆 ∈ V
110109a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑆 ∈ V)
111 restabs 23189 . . . . . . . . 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 1465 . . . . . . . 8 ((𝜑𝑎𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113112oveq1d 7446 . . . . . . 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 6909 . . . . . 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 2842 . . . . 5 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116 resttop 23184 . . . . . . . 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 3981 . . . . . . . . . 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 24818 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122121blopn 24529 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
12355, 14, 58, 122mp3an12i 1464 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124 elrestr 17475 . . . . . . . . . 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 1464 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126120, 125eqeltrrd 2840 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127 isopn3i 23106 . . . . . . . 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 2842 . . . . . 6 ((𝜑𝑎𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
13090adantr 480 . . . . . 6 ((𝜑𝑎𝑆) → 𝐹:𝑆⟶ℂ)
131101restuni 23186 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
132100, 11, 131mp2an 692 . . . . . . 7 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
133132, 101cnprest 23313 . . . . . 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 839 . . . . 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 3144 . . 3 (𝜑 → ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137 resttopon 23185 . . . . 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 23304 . . . 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 2735 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
14393, 142, 96cncfcn 24950 . . 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 2850 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cin 3962  wss 3963  ifcif 4531   cuni 4912   class class class wbr 5148  cmpt 5231  ccnv 5688  dom cdm 5689  cres 5691  cima 5692  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290  *cxr 11292   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  2c2 12319  0cn0 12524  +crp 13032  [,)cico 13386  [,]cicc 13387  seqcseq 14039  cexp 14099  abscabs 15270  cli 15517  Σcsu 15719  t crest 17467  TopOpenctopn 17468  ∞Metcxmet 21367  ballcbl 21369  fldccnfld 21382  Topctop 22915  TopOnctopon 22932  intcnt 23041   Cn ccn 23248   CnP ccnp 23249  cnccncf 24916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-ntr 23044  df-cn 23251  df-cnp 23252  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-ulm 26435
This theorem is referenced by:  pserdvlem2  26487  pserdv  26488  abelth  26500  logtayl  26717
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