MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psercn Structured version   Visualization version   GIF version

Theorem psercn 25000
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 15024 . . . . . 6 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
21rgenw 3137 . . . . 5 𝑦𝑆 Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗) ∈ V
3 pserf.f . . . . . 6 𝐹 = (𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
43fnmpt 6464 . . . . 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 5925 . . . . . . . . . . . 12 (abs “ (0[,)𝑅)) ⊆ dom abs
8 absf 14677 . . . . . . . . . . . . 13 abs:ℂ⟶ℝ
98fdmi 6500 . . . . . . . . . . . 12 dom abs = ℂ
107, 9sseqtri 3982 . . . . . . . . . . 11 (abs “ (0[,)𝑅)) ⊆ ℂ
116, 10eqsstri 3980 . . . . . . . . . 10 𝑆 ⊆ ℂ
1211a1i 11 . . . . . . . . 9 (𝜑𝑆 ⊆ ℂ)
1312sselda 3946 . . . . . . . 8 ((𝜑𝑎𝑆) → 𝑎 ∈ ℂ)
14 0cn 10611 . . . . . . . . . . 11 0 ∈ ℂ
15 eqid 2820 . . . . . . . . . . . 12 (abs ∘ − ) = (abs ∘ − )
1615cnmetdval 23355 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
1714, 13, 16sylancr 589 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘(0 − 𝑎)))
18 abssub 14666 . . . . . . . . . . 11 ((0 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
1914, 13, 18sylancr 589 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(0 − 𝑎)) = (abs‘(𝑎 − 0)))
2013subid1d 10964 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑎 − 0) = 𝑎)
2120fveq2d 6650 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (abs‘(𝑎 − 0)) = (abs‘𝑎))
2217, 19, 213eqtrd 2859 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) = (abs‘𝑎))
23 breq2 5046 . . . . . . . . . . 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 5046 . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝑆) → 𝑎𝑆)
2625, 6eleqtrdi 2921 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑎 ∈ (abs “ (0[,)𝑅)))
27 ffn 6490 . . . . . . . . . . . . . . . . . 18 (abs:ℂ⟶ℝ → abs Fn ℂ)
28 elpreima 6804 . . . . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → (𝑎 ∈ ℂ ∧ (abs‘𝑎) ∈ (0[,)𝑅)))
3130simprd 498 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ (0[,)𝑅))
32 0re 10621 . . . . . . . . . . . . . . . 16 0 ∈ ℝ
33 iccssxr 12799 . . . . . . . . . . . . . . . . 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 24991 . . . . . . . . . . . . . . . . . 18 (𝜑𝑅 ∈ (0[,]+∞))
3837adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝑆) → 𝑅 ∈ (0[,]+∞))
3933, 38sseldi 3944 . . . . . . . . . . . . . . . 16 ((𝜑𝑎𝑆) → 𝑅 ∈ ℝ*)
40 elico2 12780 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
4132, 39, 40sylancr 589 . . . . . . . . . . . . . . 15 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ (0[,)𝑅) ↔ ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅)))
4231, 41mpbid 234 . . . . . . . . . . . . . 14 ((𝜑𝑎𝑆) → ((abs‘𝑎) ∈ ℝ ∧ 0 ≤ (abs‘𝑎) ∧ (abs‘𝑎) < 𝑅))
4342simp3d 1140 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑅)
4443adantr 483 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < 𝑅)
4513abscld 14776 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (abs‘𝑎) ∈ ℝ)
46 avglt1 11854 . . . . . . . . . . . . 13 (((abs‘𝑎) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
4745, 46sylan 582 . . . . . . . . . . . 12 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → ((abs‘𝑎) < 𝑅 ↔ (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2)))
4844, 47mpbid 234 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ 𝑅 ∈ ℝ) → (abs‘𝑎) < (((abs‘𝑎) + 𝑅) / 2))
4945ltp1d 11548 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5049adantr 483 . . . . . . . . . . 11 (((𝜑𝑎𝑆) ∧ ¬ 𝑅 ∈ ℝ) → (abs‘𝑎) < ((abs‘𝑎) + 1))
5123, 24, 48, 50ifbothda 4480 . . . . . . . . . 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 5084 . . . . . . . . 9 ((𝜑𝑎𝑆) → (abs‘𝑎) < 𝑀)
5422, 53eqbrtrd 5064 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(abs ∘ − )𝑎) < 𝑀)
55 cnxmet 23357 . . . . . . . . 9 (abs ∘ − ) ∈ (∞Met‘ℂ)
5634, 3, 35, 36, 6, 52psercnlem1 24999 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (𝑀 ∈ ℝ+ ∧ (abs‘𝑎) < 𝑀𝑀 < 𝑅))
5756simp1d 1138 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ+)
5857rpxrd 12411 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ*)
59 elbl 22974 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ↔ (𝑎 ∈ ℂ ∧ (0(abs ∘ − )𝑎) < 𝑀)))
6055, 14, 58, 59mp3an12i 1461 . . . . . . . 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 6666 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) = (𝐹𝑎))
633reseq1i 5825 . . . . . . . . . 10 (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀))
6434, 3, 35, 36, 6, 56psercnlem2 24998 . . . . . . . . . . . . 13 ((𝜑𝑎𝑆) → (𝑎 ∈ (0(ball‘(abs ∘ − ))𝑀) ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)) ∧ (abs “ (0[,]𝑀)) ⊆ 𝑆))
6564simp2d 1139 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ (abs “ (0[,]𝑀)))
6664simp3d 1140 . . . . . . . . . . . 12 ((𝜑𝑎𝑆) → (abs “ (0[,]𝑀)) ⊆ 𝑆)
6765, 66sstrd 3956 . . . . . . . . . . 11 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆)
6867resmptd 5884 . . . . . . . . . 10 ((𝜑𝑎𝑆) → ((𝑦𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
6963, 68syl5eq 2867 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)))
70 eqid 2820 . . . . . . . . . 10 (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗))
7135adantr 483 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝐴:ℕ0⟶ℂ)
72 fveq2 6646 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → (𝐺𝑘) = (𝐺𝑦))
7372seqeq3d 13361 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 → seq0( + , (𝐺𝑘)) = seq0( + , (𝐺𝑦)))
7473fveq1d 6648 . . . . . . . . . . . . 13 (𝑘 = 𝑦 → (seq0( + , (𝐺𝑘))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑠))
7574cbvmptv 5145 . . . . . . . . . . . 12 (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠))
76 fveq2 6646 . . . . . . . . . . . . 13 (𝑠 = 𝑖 → (seq0( + , (𝐺𝑦))‘𝑠) = (seq0( + , (𝐺𝑦))‘𝑖))
7776mpteq2dv 5138 . . . . . . . . . . . 12 (𝑠 = 𝑖 → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
7875, 77syl5eq 2867 . . . . . . . . . . 11 (𝑠 = 𝑖 → (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠)) = (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
7978cbvmptv 5145 . . . . . . . . . 10 (𝑠 ∈ ℕ0 ↦ (𝑘 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑘))‘𝑠))) = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ (seq0( + , (𝐺𝑦))‘𝑖)))
8057rpred 12410 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 ∈ ℝ)
8156simp3d 1140 . . . . . . . . . 10 ((𝜑𝑎𝑆) → 𝑀 < 𝑅)
8234, 70, 71, 36, 79, 80, 81, 65psercn2 24997 . . . . . . . . 9 ((𝜑𝑎𝑆) → (𝑦 ∈ (0(ball‘(abs ∘ − ))𝑀) ↦ Σ𝑗 ∈ ℕ0 ((𝐺𝑦)‘𝑗)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
8369, 82eqeltrd 2911 . . . . . . . 8 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ))
84 cncff 23477 . . . . . . . 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, 61ffvelrnd 6828 . . . . . 6 ((𝜑𝑎𝑆) → ((𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀))‘𝑎) ∈ ℂ)
8762, 86eqeltrrd 2912 . . . . 5 ((𝜑𝑎𝑆) → (𝐹𝑎) ∈ ℂ)
8887ralrimiva 3169 . . . 4 (𝜑 → ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ)
89 ffnfv 6858 . . . 4 (𝐹:𝑆⟶ℂ ↔ (𝐹 Fn 𝑆 ∧ ∀𝑎𝑆 (𝐹𝑎) ∈ ℂ))
905, 88, 89sylanbrc 585 . . 3 (𝜑𝐹:𝑆⟶ℂ)
9167, 11sstrdi 3958 . . . . . . . . 9 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ)
92 ssid 3968 . . . . . . . . 9 ℂ ⊆ ℂ
93 eqid 2820 . . . . . . . . . 10 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
94 eqid 2820 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
9593cnfldtopon 23367 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9695toponrestid 21505 . . . . . . . . . 10 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
9793, 94, 96cncfcn 23494 . . . . . . . . 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 588 . . . . . . . 8 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
9983, 98eleqtrd 2913 . . . . . . 7 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) Cn (TopOpen‘ℂfld)))
10093cnfldtop 23368 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ Top
101 unicntop 23370 . . . . . . . . . 10 ℂ = (TopOpen‘ℂfld)
102101restuni 21746 . . . . . . . . 9 (((TopOpen‘ℂfld) ∈ Top ∧ (0(ball‘(abs ∘ − ))𝑀) ⊆ ℂ) → (0(ball‘(abs ∘ − ))𝑀) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
103100, 91, 102sylancr 589 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
10461, 103eleqtrd 2913 . . . . . . 7 ((𝜑𝑎𝑆) → 𝑎 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
105 eqid 2820 . . . . . . . 8 ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀))
106105cncnpi 21862 . . . . . . 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 586 . . . . . 6 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ ((((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
108 cnex 10596 . . . . . . . . . . 11 ℂ ∈ V
109108, 11ssexi 5202 . . . . . . . . . 10 𝑆 ∈ V
110109a1i 11 . . . . . . . . 9 ((𝜑𝑎𝑆) → 𝑆 ∈ V)
111 restabs 21749 . . . . . . . . 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 1462 . . . . . . . 8 ((𝜑𝑎𝑆) → (((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) = ((TopOpen‘ℂfld) ↾t (0(ball‘(abs ∘ − ))𝑀)))
113112oveq1d 7148 . . . . . . 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 6648 . . . . . 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 2914 . . . . 5 ((𝜑𝑎𝑆) → (𝐹 ↾ (0(ball‘(abs ∘ − ))𝑀)) ∈ (((((TopOpen‘ℂfld) ↾t 𝑆) ↾t (0(ball‘(abs ∘ − ))𝑀)) CnP (TopOpen‘ℂfld))‘𝑎))
116 resttop 21744 . . . . . . . 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 3930 . . . . . . . . . 10 ((0(ball‘(abs ∘ − ))𝑀) ⊆ 𝑆 ↔ ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12067, 119sylib 220 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) = (0(ball‘(abs ∘ − ))𝑀))
12193cnfldtopn 23366 . . . . . . . . . . . 12 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
122121blopn 23086 . . . . . . . . . . 11 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑀 ∈ ℝ*) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
12355, 14, 58, 122mp3an12i 1461 . . . . . . . . . 10 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ (TopOpen‘ℂfld))
124 elrestr 16681 . . . . . . . . . 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 1461 . . . . . . . . 9 ((𝜑𝑎𝑆) → ((0(ball‘(abs ∘ − ))𝑀) ∩ 𝑆) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
126120, 125eqeltrrd 2912 . . . . . . . 8 ((𝜑𝑎𝑆) → (0(ball‘(abs ∘ − ))𝑀) ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
127 isopn3i 21666 . . . . . . . 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 589 . . . . . . 7 ((𝜑𝑎𝑆) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)) = (0(ball‘(abs ∘ − ))𝑀))
12961, 128eleqtrrd 2914 . . . . . 6 ((𝜑𝑎𝑆) → 𝑎 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘(0(ball‘(abs ∘ − ))𝑀)))
13090adantr 483 . . . . . 6 ((𝜑𝑎𝑆) → 𝐹:𝑆⟶ℂ)
131101restuni 21746 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ⊆ ℂ) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
132100, 11, 131mp2an 690 . . . . . . 7 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆)
133132, 101cnprest 21873 . . . . . 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 836 . . . . 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 259 . . . 4 ((𝜑𝑎𝑆) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
136135ralrimiva 3169 . . 3 (𝜑 → ∀𝑎𝑆 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t 𝑆) CnP (TopOpen‘ℂfld))‘𝑎))
137 resttopon 21745 . . . . 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 21864 . . . 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 585 . 2 (𝜑𝐹 ∈ (((TopOpen‘ℂfld) ↾t 𝑆) Cn (TopOpen‘ℂfld)))
142 eqid 2820 . . . 4 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
14393, 142, 96cncfcn 23494 . . 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 2922 1 (𝜑𝐹 ∈ (𝑆cn→ℂ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  {crab 3129  Vcvv 3473  cin 3912  wss 3913  ifcif 4443   cuni 4814   class class class wbr 5042  cmpt 5122  ccnv 5530  dom cdm 5531  cres 5533  cima 5534  ccom 5535   Fn wfn 6326  wf 6327  cfv 6331  (class class class)co 7133  supcsup 8882  cc 10513  cr 10514  0cc0 10515  1c1 10516   + caddc 10518   · cmul 10520  +∞cpnf 10650  *cxr 10652   < clt 10653  cle 10654  cmin 10848   / cdiv 11275  2c2 11671  0cn0 11876  +crp 12368  [,)cico 12719  [,]cicc 12720  seqcseq 13353  cexp 13414  abscabs 14573  cli 14821  Σcsu 15022  t crest 16673  TopOpenctopn 16674  ∞Metcxmet 20506  ballcbl 20508  fldccnfld 20521  Topctop 21477  TopOnctopon 21494  intcnt 21601   Cn ccn 21808   CnP ccnp 21809  cnccncf 23460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593  ax-addf 10594  ax-mulf 10595
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-pm 8387  df-ixp 8440  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-fsupp 8812  df-fi 8853  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-q 12328  df-rp 12369  df-xneg 12486  df-xadd 12487  df-xmul 12488  df-ico 12723  df-icc 12724  df-fz 12877  df-fzo 13018  df-fl 13146  df-seq 13354  df-exp 13415  df-hash 13676  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-limsup 14808  df-clim 14825  df-rlim 14826  df-sum 15023  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-mulr 16558  df-starv 16559  df-sca 16560  df-vsca 16561  df-ip 16562  df-tset 16563  df-ple 16564  df-ds 16566  df-unif 16567  df-hom 16568  df-cco 16569  df-rest 16675  df-topn 16676  df-0g 16694  df-gsum 16695  df-topgen 16696  df-pt 16697  df-prds 16700  df-xrs 16754  df-qtop 16759  df-imas 16760  df-xps 16762  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-mulg 18204  df-cntz 18426  df-cmn 18887  df-psmet 20513  df-xmet 20514  df-met 20515  df-bl 20516  df-mopn 20517  df-cnfld 20522  df-top 21478  df-topon 21495  df-topsp 21517  df-bases 21530  df-ntr 21604  df-cn 21811  df-cnp 21812  df-tx 22146  df-hmeo 22339  df-xms 22906  df-ms 22907  df-tms 22908  df-cncf 23462  df-ulm 24951
This theorem is referenced by:  pserdvlem2  25002  pserdv  25003  abelth  25015  logtayl  25230
  Copyright terms: Public domain W3C validator