Step | Hyp | Ref
| Expression |
1 | | dvply1.f |
. . 3
⊢ (𝜑 → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
2 | 1 | oveq2d 6666 |
. 2
⊢ (𝜑 → (ℂ D 𝐹) = (ℂ D (𝑧 ∈ ℂ ↦
Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘))))) |
3 | | eqid 2622 |
. . . . . 6
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
4 | 3 | cnfldtop 22587 |
. . . . 5
⊢
(TopOpen‘ℂfld) ∈ Top |
5 | 3 | cnfldtopon 22586 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
6 | 5 | toponunii 20721 |
. . . . . 6
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
7 | 6 | restid 16094 |
. . . . 5
⊢
((TopOpen‘ℂfld) ∈ Top →
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld)) |
8 | 4, 7 | ax-mp 5 |
. . . 4
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld) |
9 | 8 | eqcomi 2631 |
. . 3
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
10 | | cnelprrecn 10029 |
. . . 4
⊢ ℂ
∈ {ℝ, ℂ} |
11 | 10 | a1i 11 |
. . 3
⊢ (𝜑 → ℂ ∈ {ℝ,
ℂ}) |
12 | 6 | topopn 20711 |
. . . 4
⊢
((TopOpen‘ℂfld) ∈ Top → ℂ ∈
(TopOpen‘ℂfld)) |
13 | 4, 12 | mp1i 13 |
. . 3
⊢ (𝜑 → ℂ ∈
(TopOpen‘ℂfld)) |
14 | | fzfid 12772 |
. . 3
⊢ (𝜑 → (0...𝑁) ∈ Fin) |
15 | | dvply1.a |
. . . . . . 7
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
16 | | elfznn0 12433 |
. . . . . . 7
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) |
17 | | ffvelrn 6357 |
. . . . . . 7
⊢ ((𝐴:ℕ0⟶ℂ ∧
𝑘 ∈
ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
18 | 15, 16, 17 | syl2an 494 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
19 | 18 | adantr 481 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ) |
20 | | simpr 477 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑧 ∈ ℂ) |
21 | 16 | ad2antlr 763 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → 𝑘 ∈ ℕ0) |
22 | 20, 21 | expcld 13008 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → (𝑧↑𝑘) ∈ ℂ) |
23 | 19, 22 | mulcld 10060 |
. . . 4
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ) |
24 | 23 | 3impa 1259 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · (𝑧↑𝑘)) ∈ ℂ) |
25 | 18 | 3adant3 1081 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → (𝐴‘𝑘) ∈ ℂ) |
26 | | 0cnd 10033 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 = 0) → 0 ∈
ℂ) |
27 | | simpl2 1065 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ (0...𝑁)) |
28 | 27, 16 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ0) |
29 | 28 | nn0cnd 11353 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℂ) |
30 | | simpl3 1066 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑧 ∈ ℂ) |
31 | | simpr 477 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → ¬ 𝑘 = 0) |
32 | | elnn0 11294 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
↔ (𝑘 ∈ ℕ
∨ 𝑘 =
0)) |
33 | 28, 32 | sylib 208 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 ∈ ℕ ∨ 𝑘 = 0)) |
34 | | orel2 398 |
. . . . . . . . 9
⊢ (¬
𝑘 = 0 → ((𝑘 ∈ ℕ ∨ 𝑘 = 0) → 𝑘 ∈ ℕ)) |
35 | 31, 33, 34 | sylc 65 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ) |
36 | | nnm1nn0 11334 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈
ℕ0) |
37 | 35, 36 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 − 1) ∈
ℕ0) |
38 | 30, 37 | expcld 13008 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑧↑(𝑘 − 1)) ∈ ℂ) |
39 | 29, 38 | mulcld 10060 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈
ℂ) |
40 | 26, 39 | ifclda 4120 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈
ℂ) |
41 | 25, 40 | mulcld 10060 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁) ∧ 𝑧 ∈ ℂ) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈
ℂ) |
42 | 10 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ℂ ∈ {ℝ,
ℂ}) |
43 | | c0ex 10034 |
. . . . . 6
⊢ 0 ∈
V |
44 | | ovex 6678 |
. . . . . 6
⊢ (𝑘 · (𝑧↑(𝑘 − 1))) ∈ V |
45 | 43, 44 | ifex 4156 |
. . . . 5
⊢ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ V |
46 | 45 | a1i 11 |
. . . 4
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑧 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈ V) |
47 | 16 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
48 | | dvexp2 23717 |
. . . . 5
⊢ (𝑘 ∈ ℕ0
→ (ℂ D (𝑧 ∈
ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) |
49 | 47, 48 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (ℂ D (𝑧 ∈ ℂ ↦ (𝑧↑𝑘))) = (𝑧 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) |
50 | 42, 22, 46, 49, 18 | dvmptcmul 23727 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (ℂ D (𝑧 ∈ ℂ ↦ ((𝐴‘𝑘) · (𝑧↑𝑘)))) = (𝑧 ∈ ℂ ↦ ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))))) |
51 | 9, 3, 11, 13, 14, 24, 41, 50 | dvmptfsum 23738 |
. 2
⊢ (𝜑 → (ℂ D (𝑧 ∈ ℂ ↦
Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑧↑𝑘)))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))))) |
52 | | elfznn 12370 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) |
53 | 52 | nnne0d 11065 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0) |
54 | 53 | neneqd 2799 |
. . . . . . . . 9
⊢ (𝑘 ∈ (1...𝑁) → ¬ 𝑘 = 0) |
55 | 54 | adantl 482 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0) |
56 | 55 | iffalsed 4097 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1)))) |
57 | 56 | oveq2d 6666 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1))))) |
58 | 57 | sumeq2dv 14433 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1))))) |
59 | | 1eluzge0 11732 |
. . . . . . 7
⊢ 1 ∈
(ℤ≥‘0) |
60 | | fzss1 12380 |
. . . . . . 7
⊢ (1 ∈
(ℤ≥‘0) → (1...𝑁) ⊆ (0...𝑁)) |
61 | 59, 60 | mp1i 13 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (1...𝑁) ⊆ (0...𝑁)) |
62 | 15 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝐴:ℕ0⟶ℂ) |
63 | 52 | nnnn0d 11351 |
. . . . . . . 8
⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ0) |
64 | 62, 63, 17 | syl2an 494 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
65 | 53 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0) |
66 | 65 | neneqd 2799 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ¬ 𝑘 = 0) |
67 | 66 | iffalsed 4097 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = (𝑘 · (𝑧↑(𝑘 − 1)))) |
68 | 63 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℕ0) |
69 | 68 | nn0cnd 11353 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ) |
70 | | simplr 792 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → 𝑧 ∈ ℂ) |
71 | 52, 36 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (1...𝑁) → (𝑘 − 1) ∈
ℕ0) |
72 | 71 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 − 1) ∈
ℕ0) |
73 | 70, 72 | expcld 13008 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑧↑(𝑘 − 1)) ∈ ℂ) |
74 | 69, 73 | mulcld 10060 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · (𝑧↑(𝑘 − 1))) ∈
ℂ) |
75 | 67, 74 | eqeltrd 2701 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) ∈
ℂ) |
76 | 64, 75 | mulcld 10060 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) ∈
ℂ) |
77 | | eldifn 3733 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ (1...𝑁)) |
78 | | 0p1e1 11132 |
. . . . . . . . . . . . . 14
⊢ (0 + 1) =
1 |
79 | 78 | oveq1i 6660 |
. . . . . . . . . . . . 13
⊢ ((0 +
1)...𝑁) = (1...𝑁) |
80 | 79 | eleq2i 2693 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0 + 1)...𝑁) ↔ 𝑘 ∈ (1...𝑁)) |
81 | 77, 80 | sylnibr 319 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → ¬ 𝑘 ∈ ((0 + 1)...𝑁)) |
82 | 81 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ¬ 𝑘 ∈ ((0 + 1)...𝑁)) |
83 | | eldifi 3732 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...𝑁) ∖ (1...𝑁)) → 𝑘 ∈ (0...𝑁)) |
84 | 83 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 ∈ (0...𝑁)) |
85 | | dvply1.n |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
86 | | nn0uz 11722 |
. . . . . . . . . . . . . 14
⊢
ℕ0 = (ℤ≥‘0) |
87 | 85, 86 | syl6eleq 2711 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘0)) |
88 | 87 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑁 ∈
(ℤ≥‘0)) |
89 | | elfzp12 12419 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈
(ℤ≥‘0) → (𝑘 ∈ (0...𝑁) ↔ (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)))) |
90 | 88, 89 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 ∈ (0...𝑁) ↔ (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)))) |
91 | 84, 90 | mpbid 222 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → (𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁))) |
92 | | orel2 398 |
. . . . . . . . . 10
⊢ (¬
𝑘 ∈ ((0 + 1)...𝑁) → ((𝑘 = 0 ∨ 𝑘 ∈ ((0 + 1)...𝑁)) → 𝑘 = 0)) |
93 | 82, 91, 92 | sylc 65 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → 𝑘 = 0) |
94 | 93 | iftrued 4094 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))) = 0) |
95 | 94 | oveq2d 6666 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = ((𝐴‘𝑘) · 0)) |
96 | 62, 16, 17 | syl2an 494 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (𝐴‘𝑘) ∈ ℂ) |
97 | 96 | mul01d 10235 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · 0) = 0) |
98 | 83, 97 | sylan2 491 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · 0) = 0) |
99 | 95, 98 | eqtrd 2656 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ ((0...𝑁) ∖ (1...𝑁))) → ((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = 0) |
100 | | fzfid 12772 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (0...𝑁) ∈ Fin) |
101 | 61, 76, 99, 100 | fsumss 14456 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) |
102 | | elfznn0 12433 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → 𝑗 ∈ ℕ0) |
103 | 102 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℕ0) |
104 | 103 | nn0cnd 11353 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑗 ∈ ℂ) |
105 | | ax-1cn 9994 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℂ |
106 | | pncan 10287 |
. . . . . . . . . . . . 13
⊢ ((𝑗 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑗 + 1)
− 1) = 𝑗) |
107 | 104, 105,
106 | sylancl 694 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) − 1) = 𝑗) |
108 | 107 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑((𝑗 + 1) − 1)) = (𝑧↑𝑗)) |
109 | 108 | oveq2d 6666 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))) = ((𝑗 + 1) · (𝑧↑𝑗))) |
110 | 109 | oveq2d 6666 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑𝑗)))) |
111 | 15 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝐴:ℕ0⟶ℂ) |
112 | | peano2nn0 11333 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
ℕ0) |
113 | 102, 112 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ (0...(𝑁 − 1)) → (𝑗 + 1) ∈
ℕ0) |
114 | 113 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈
ℕ0) |
115 | 111, 114 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝐴‘(𝑗 + 1)) ∈ ℂ) |
116 | 114 | nn0cnd 11353 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑗 + 1) ∈ ℂ) |
117 | | simplr 792 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → 𝑧 ∈ ℂ) |
118 | 117, 103 | expcld 13008 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (𝑧↑𝑗) ∈ ℂ) |
119 | 115, 116,
118 | mulassd 10063 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) · (𝑧↑𝑗)) = ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑𝑗)))) |
120 | 115, 116 | mulcomd 10061 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1)))) |
121 | 120 | oveq1d 6665 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → (((𝐴‘(𝑗 + 1)) · (𝑗 + 1)) · (𝑧↑𝑗)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗))) |
122 | 110, 119,
121 | 3eqtr2d 2662 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑗 ∈ (0...(𝑁 − 1))) → ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗))) |
123 | 122 | sumeq2dv 14433 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑗 ∈ (0...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = Σ𝑗 ∈ (0...(𝑁 − 1))(((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗))) |
124 | | 1m1e0 11089 |
. . . . . . . . 9
⊢ (1
− 1) = 0 |
125 | 124 | oveq1i 6660 |
. . . . . . . 8
⊢ ((1
− 1)...(𝑁 − 1))
= (0...(𝑁 −
1)) |
126 | 125 | sumeq1i 14428 |
. . . . . . 7
⊢
Σ𝑗 ∈ ((1
− 1)...(𝑁 −
1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = Σ𝑗 ∈ (0...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) |
127 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝑘 + 1) = (𝑗 + 1)) |
128 | 127 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑗 → (𝐴‘(𝑘 + 1)) = (𝐴‘(𝑗 + 1))) |
129 | 127, 128 | oveq12d 6668 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) = ((𝑗 + 1) · (𝐴‘(𝑗 + 1)))) |
130 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑘 = 𝑗 → (𝑧↑𝑘) = (𝑧↑𝑗)) |
131 | 129, 130 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑘 = 𝑗 → (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = (((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗))) |
132 | 131 | cbvsumv 14426 |
. . . . . . 7
⊢
Σ𝑘 ∈
(0...(𝑁 − 1))(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘)) = Σ𝑗 ∈ (0...(𝑁 − 1))(((𝑗 + 1) · (𝐴‘(𝑗 + 1))) · (𝑧↑𝑗)) |
133 | 123, 126,
132 | 3eqtr4g 2681 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑗 ∈ ((1 − 1)...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) = Σ𝑘 ∈ (0...(𝑁 − 1))(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘))) |
134 | | 1zzd 11408 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 1 ∈
ℤ) |
135 | 85 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈
ℕ0) |
136 | 135 | nn0zd 11480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 𝑁 ∈ ℤ) |
137 | 64, 74 | mulcld 10060 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) ∈
ℂ) |
138 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑘 = (𝑗 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑗 + 1))) |
139 | | id 22 |
. . . . . . . . 9
⊢ (𝑘 = (𝑗 + 1) → 𝑘 = (𝑗 + 1)) |
140 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 + 1) → (𝑘 − 1) = ((𝑗 + 1) − 1)) |
141 | 140 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑘 = (𝑗 + 1) → (𝑧↑(𝑘 − 1)) = (𝑧↑((𝑗 + 1) − 1))) |
142 | 139, 141 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑘 = (𝑗 + 1) → (𝑘 · (𝑧↑(𝑘 − 1))) = ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1)))) |
143 | 138, 142 | oveq12d 6668 |
. . . . . . 7
⊢ (𝑘 = (𝑗 + 1) → ((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) = ((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))) |
144 | 134, 134,
136, 137, 143 | fsumshftm 14513 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) = Σ𝑗 ∈ ((1 − 1)...(𝑁 − 1))((𝐴‘(𝑗 + 1)) · ((𝑗 + 1) · (𝑧↑((𝑗 + 1) − 1))))) |
145 | | elfznn0 12433 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0) |
146 | 145 | adantl 482 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 𝑘 ∈ ℕ0) |
147 | | ovex 6678 |
. . . . . . . . 9
⊢ ((𝑘 + 1) · (𝐴‘(𝑘 + 1))) ∈ V |
148 | | dvply1.b |
. . . . . . . . . 10
⊢ 𝐵 = (𝑘 ∈ ℕ0 ↦ ((𝑘 + 1) · (𝐴‘(𝑘 + 1)))) |
149 | 148 | fvmpt2 6291 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ0
∧ ((𝑘 + 1) ·
(𝐴‘(𝑘 + 1))) ∈ V) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1)))) |
150 | 146, 147,
149 | sylancl 694 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐵‘𝑘) = ((𝑘 + 1) · (𝐴‘(𝑘 + 1)))) |
151 | 150 | oveq1d 6665 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐵‘𝑘) · (𝑧↑𝑘)) = (((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘))) |
152 | 151 | sumeq2dv 14433 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)) = Σ𝑘 ∈ (0...(𝑁 − 1))(((𝑘 + 1) · (𝐴‘(𝑘 + 1))) · (𝑧↑𝑘))) |
153 | 133, 144,
152 | 3eqtr4d 2666 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (1...𝑁)((𝐴‘𝑘) · (𝑘 · (𝑧↑(𝑘 − 1)))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘))) |
154 | 58, 101, 153 | 3eqtr3d 2664 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1))))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘))) |
155 | 154 | mpteq2dva 4744 |
. . 3
⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)))) |
156 | | dvply1.g |
. . 3
⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 − 1))((𝐵‘𝑘) · (𝑧↑𝑘)))) |
157 | 155, 156 | eqtr4d 2659 |
. 2
⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · if(𝑘 = 0, 0, (𝑘 · (𝑧↑(𝑘 − 1)))))) = 𝐺) |
158 | 2, 51, 157 | 3eqtrd 2660 |
1
⊢ (𝜑 → (ℂ D 𝐹) = 𝐺) |