Step | Hyp | Ref
| Expression |
1 | | plyssc 23956 |
. . 3
⊢
(Poly‘𝑆)
⊆ (Poly‘ℂ) |
2 | | vieta1.4 |
. . 3
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
3 | 1, 2 | sseldi 3601 |
. 2
⊢ (𝜑 → 𝐹 ∈
(Poly‘ℂ)) |
4 | | vieta1.6 |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℕ) |
5 | | eqeq1 2626 |
. . . . . . 7
⊢ (𝑦 = 1 → (𝑦 = (deg‘𝑓) ↔ 1 = (deg‘𝑓))) |
6 | 5 | anbi1d 741 |
. . . . . 6
⊢ (𝑦 = 1 → ((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (1 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)))) |
7 | 6 | imbi1d 331 |
. . . . 5
⊢ (𝑦 = 1 → (((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((1 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
8 | 7 | ralbidv 2986 |
. . . 4
⊢ (𝑦 = 1 → (∀𝑓 ∈
(Poly‘ℂ)((𝑦 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((1
= (deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
9 | | eqeq1 2626 |
. . . . . . 7
⊢ (𝑦 = 𝑑 → (𝑦 = (deg‘𝑓) ↔ 𝑑 = (deg‘𝑓))) |
10 | 9 | anbi1d 741 |
. . . . . 6
⊢ (𝑦 = 𝑑 → ((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑑 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)))) |
11 | 10 | imbi1d 331 |
. . . . 5
⊢ (𝑦 = 𝑑 → (((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑑 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
12 | 11 | ralbidv 2986 |
. . . 4
⊢ (𝑦 = 𝑑 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
13 | | eqeq1 2626 |
. . . . . . 7
⊢ (𝑦 = (𝑑 + 1) → (𝑦 = (deg‘𝑓) ↔ (𝑑 + 1) = (deg‘𝑓))) |
14 | 13 | anbi1d 741 |
. . . . . 6
⊢ (𝑦 = (𝑑 + 1) → ((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)))) |
15 | 14 | imbi1d 331 |
. . . . 5
⊢ (𝑦 = (𝑑 + 1) → (((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ (((𝑑 + 1) = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
16 | 15 | ralbidv 2986 |
. . . 4
⊢ (𝑦 = (𝑑 + 1) → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈
(Poly‘ℂ)(((𝑑 +
1) = (deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
17 | | eqeq1 2626 |
. . . . . . 7
⊢ (𝑦 = 𝑁 → (𝑦 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝑓))) |
18 | 17 | anbi1d 741 |
. . . . . 6
⊢ (𝑦 = 𝑁 → ((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)))) |
19 | 18 | imbi1d 331 |
. . . . 5
⊢ (𝑦 = 𝑁 → (((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑁 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
20 | 19 | ralbidv 2986 |
. . . 4
⊢ (𝑦 = 𝑁 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈
(Poly‘ℂ)((𝑁 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
21 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(coeff‘𝑓) =
(coeff‘𝑓) |
22 | 21 | coef3 23988 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ (Poly‘ℂ)
→ (coeff‘𝑓):ℕ0⟶ℂ) |
23 | 22 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (coeff‘𝑓):ℕ0⟶ℂ) |
24 | | 0nn0 11307 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℕ0 |
25 | | ffvelrn 6357 |
. . . . . . . . . . . 12
⊢
(((coeff‘𝑓):ℕ0⟶ℂ ∧ 0
∈ ℕ0) → ((coeff‘𝑓)‘0) ∈ ℂ) |
26 | 23, 24, 25 | sylancl 694 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((coeff‘𝑓)‘0) ∈ ℂ) |
27 | | 1nn0 11308 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ0 |
28 | | ffvelrn 6357 |
. . . . . . . . . . . 12
⊢
(((coeff‘𝑓):ℕ0⟶ℂ ∧ 1
∈ ℕ0) → ((coeff‘𝑓)‘1) ∈ ℂ) |
29 | 23, 27, 28 | sylancl 694 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((coeff‘𝑓)‘1) ∈ ℂ) |
30 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ 1 = (deg‘𝑓)) |
31 | 30 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((coeff‘𝑓)‘1) = ((coeff‘𝑓)‘(deg‘𝑓))) |
32 | | ax-1ne0 10005 |
. . . . . . . . . . . . . . . 16
⊢ 1 ≠
0 |
33 | 32 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ 1 ≠ 0) |
34 | 30, 33 | eqnetrrd 2862 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (deg‘𝑓) ≠
0) |
35 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = 0𝑝 →
(deg‘𝑓) =
(deg‘0𝑝)) |
36 | | dgr0 24018 |
. . . . . . . . . . . . . . . 16
⊢
(deg‘0𝑝) = 0 |
37 | 35, 36 | syl6eq 2672 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 0𝑝 →
(deg‘𝑓) =
0) |
38 | 37 | necon3i 2826 |
. . . . . . . . . . . . . 14
⊢
((deg‘𝑓) ≠
0 → 𝑓 ≠
0𝑝) |
39 | 34, 38 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ 𝑓 ≠
0𝑝) |
40 | | eqid 2622 |
. . . . . . . . . . . . . . . 16
⊢
(deg‘𝑓) =
(deg‘𝑓) |
41 | 40, 21 | dgreq0 24021 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ (Poly‘ℂ)
→ (𝑓 =
0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0)) |
42 | 41 | necon3bid 2838 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ (Poly‘ℂ)
→ (𝑓 ≠
0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)) |
43 | 42 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (𝑓 ≠
0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)) |
44 | 39, 43 | mpbid 222 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0) |
45 | 31, 44 | eqnetrd 2861 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((coeff‘𝑓)‘1) ≠ 0) |
46 | 26, 29, 45 | divcld 10801 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) |
47 | 46 | negcld 10379 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) |
48 | | id 22 |
. . . . . . . . . 10
⊢ (𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) → 𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) |
49 | 48 | sumsn 14475 |
. . . . . . . . 9
⊢
((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧
-(((coeff‘𝑓)‘0)
/ ((coeff‘𝑓)‘1)) ∈ ℂ) →
Σ𝑥 ∈
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) |
50 | 47, 47, 49 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ Σ𝑥 ∈
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) |
51 | 50 | adantrr 753 |
. . . . . . 7
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) |
52 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (◡𝑓 “ {0}) = (◡𝑓 “ {0}) |
53 | 52 | fta1 24063 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 𝑓 ≠
0𝑝) → ((◡𝑓 “ {0}) ∈ Fin ∧
(#‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) |
54 | 39, 53 | syldan 487 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((◡𝑓 “ {0}) ∈ Fin ∧
(#‘(◡𝑓 “ {0})) ≤ (deg‘𝑓))) |
55 | 54 | simpld 475 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (◡𝑓 “ {0}) ∈ Fin) |
56 | 55 | adantrr 753 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) → (◡𝑓 “ {0}) ∈ Fin) |
57 | 21, 40 | coeid2 23995 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘))) |
58 | 47, 57 | syldan 487 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘))) |
59 | 30 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (0...1) = (0...(deg‘𝑓))) |
60 | 59 | sumeq1d 14431 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ Σ𝑘 ∈
(0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘))) |
61 | | nn0uz 11722 |
. . . . . . . . . . . . . . 15
⊢
ℕ0 = (ℤ≥‘0) |
62 | | 1e0p1 11552 |
. . . . . . . . . . . . . . 15
⊢ 1 = (0 +
1) |
63 | | fveq2 6191 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 1 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘1)) |
64 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 1 →
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) |
65 | 63, 64 | oveq12d 6668 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 1 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘1) ·
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) |
66 | 23 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
∧ 𝑘 ∈
ℕ0) → ((coeff‘𝑓)‘𝑘) ∈ ℂ) |
67 | | expcl 12878 |
. . . . . . . . . . . . . . . . 17
⊢
((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ) |
68 | 47, 67 | sylan 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
∧ 𝑘 ∈
ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ) |
69 | 66, 68 | mulcld 10060 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
∧ 𝑘 ∈
ℕ0) → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) ∈
ℂ) |
70 | | 0z 11388 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℤ |
71 | 47 | exp0d 13002 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0) = 1) |
72 | 71 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) =
(((coeff‘𝑓)‘0)
· 1)) |
73 | 26 | mulid1d 10057 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) · 1) = ((coeff‘𝑓)‘0)) |
74 | 72, 73 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) =
((coeff‘𝑓)‘0)) |
75 | 74, 26 | eqeltrd 2701 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈
ℂ) |
76 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘0)) |
77 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 0 →
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) |
78 | 76, 77 | oveq12d 6668 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 0 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) ·
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))) |
79 | 78 | fsum1 14476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℤ ∧ (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈
ℂ) → Σ𝑘
∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) ·
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))) |
80 | 70, 75, 79 | sylancr 695 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ Σ𝑘 ∈
(0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) ·
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))) |
81 | 80, 74 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ Σ𝑘 ∈
(0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0)) |
82 | 81, 24 | jctil 560 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (0 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0))) |
83 | 47 | exp1d 13003 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1) = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) |
84 | 83 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) =
(((coeff‘𝑓)‘1)
· -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))) |
85 | 29, 46 | mulneg2d 10484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘1) · -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) =
-(((coeff‘𝑓)‘1)
· (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))) |
86 | 26, 29, 45 | divcan2d 10803 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) =
((coeff‘𝑓)‘0)) |
87 | 86 | negeqd 10275 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) =
-((coeff‘𝑓)‘0)) |
88 | 84, 85, 87 | 3eqtrd 2660 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) =
-((coeff‘𝑓)‘0)) |
89 | 88 | oveq2d 6666 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) ·
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = (((coeff‘𝑓)‘0) +
-((coeff‘𝑓)‘0))) |
90 | 26 | negidd 10382 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)) =
0) |
91 | 89, 90 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) ·
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = 0) |
92 | 61, 62, 65, 69, 82, 91 | fsump1i 14500 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (1 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0)) |
93 | 92 | simprd 479 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ Σ𝑘 ∈
(0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0) |
94 | 58, 60, 93 | 3eqtr2d 2662 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0) |
95 | | plyf 23954 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ (Poly‘ℂ)
→ 𝑓:ℂ⟶ℂ) |
96 | | ffn 6045 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℂ⟶ℂ →
𝑓 Fn
ℂ) |
97 | 95, 96 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ (Poly‘ℂ)
→ 𝑓 Fn
ℂ) |
98 | 97 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ 𝑓 Fn
ℂ) |
99 | | fniniseg 6338 |
. . . . . . . . . . . . 13
⊢ (𝑓 Fn ℂ →
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (◡𝑓 “ {0}) ↔ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧
(𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0))) |
100 | 98, 99 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (◡𝑓 “ {0}) ↔ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧
(𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0))) |
101 | 47, 94, 100 | mpbir2and 957 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (◡𝑓 “ {0})) |
102 | 101 | snssd 4340 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (◡𝑓 “ {0})) |
103 | 102 | adantrr 753 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) →
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (◡𝑓 “ {0})) |
104 | | hashsng 13159 |
. . . . . . . . . . . . . 14
⊢
(-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ →
(#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1) |
105 | 47, 104 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1) |
106 | 105, 30 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓)) |
107 | 106 | adantrr 753 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) →
(#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓)) |
108 | | simprr 796 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) → (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) |
109 | 107, 108 | eqtr4d 2659 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) →
(#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (#‘(◡𝑓 “ {0}))) |
110 | | snfi 8038 |
. . . . . . . . . . . 12
⊢
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin |
111 | | hashen 13135 |
. . . . . . . . . . . 12
⊢
(({-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin ∧ (◡𝑓 “ {0}) ∈ Fin) →
((#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (#‘(◡𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (◡𝑓 “ {0}))) |
112 | 110, 55, 111 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (#‘(◡𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (◡𝑓 “ {0}))) |
113 | 112 | adantrr 753 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) →
((#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (#‘(◡𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (◡𝑓 “ {0}))) |
114 | 109, 113 | mpbid 222 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) →
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (◡𝑓 “ {0})) |
115 | | fisseneq 8171 |
. . . . . . . . 9
⊢ (((◡𝑓 “ {0}) ∈ Fin ∧
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (◡𝑓 “ {0}) ∧ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (◡𝑓 “ {0})) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (◡𝑓 “ {0})) |
116 | 56, 103, 114, 115 | syl3anc 1326 |
. . . . . . . 8
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) →
{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (◡𝑓 “ {0})) |
117 | 116 | sumeq1d 14431 |
. . . . . . 7
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = Σ𝑥 ∈ (◡𝑓 “ {0})𝑥) |
118 | | 1m1e0 11089 |
. . . . . . . . . . . 12
⊢ (1
− 1) = 0 |
119 | 30 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (1 − 1) = ((deg‘𝑓) − 1)) |
120 | 118, 119 | syl5eqr 2670 |
. . . . . . . . . . 11
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ 0 = ((deg‘𝑓)
− 1)) |
121 | 120 | fveq2d 6195 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ ((coeff‘𝑓)‘0) = ((coeff‘𝑓)‘((deg‘𝑓) − 1))) |
122 | 121, 31 | oveq12d 6668 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) /
((coeff‘𝑓)‘(deg‘𝑓)))) |
123 | 122 | negeqd 10275 |
. . . . . . . 8
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ 1 = (deg‘𝑓))
→ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) /
((coeff‘𝑓)‘(deg‘𝑓)))) |
124 | 123 | adantrr 753 |
. . . . . . 7
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) =
-(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) |
125 | 51, 117, 124 | 3eqtr3d 2664 |
. . . . . 6
⊢ ((𝑓 ∈ (Poly‘ℂ)
∧ (1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) |
126 | 125 | ex 450 |
. . . . 5
⊢ (𝑓 ∈ (Poly‘ℂ)
→ ((1 = (deg‘𝑓)
∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
127 | 126 | rgen 2922 |
. . . 4
⊢
∀𝑓 ∈
(Poly‘ℂ)((1 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) |
128 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥) |
129 | 128 | cbvsumv 14426 |
. . . . . . . . . 10
⊢
Σ𝑦 ∈
(◡𝑓 “ {0})𝑦 = Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 |
130 | 129 | eqeq1i 2627 |
. . . . . . . . 9
⊢
(Σ𝑦 ∈
(◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) |
131 | 130 | imbi2i 326 |
. . . . . . . 8
⊢ (((𝑑 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑑 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
132 | 131 | ralbii 2980 |
. . . . . . 7
⊢
(∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
133 | | eqid 2622 |
. . . . . . . . 9
⊢
(coeff‘𝑔) =
(coeff‘𝑔) |
134 | | eqid 2622 |
. . . . . . . . 9
⊢
(deg‘𝑔) =
(deg‘𝑔) |
135 | | eqid 2622 |
. . . . . . . . 9
⊢ (◡𝑔 “ {0}) = (◡𝑔 “ {0}) |
136 | | simp1r 1086 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → 𝑔 ∈
(Poly‘ℂ)) |
137 | | simp3r 1090 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → (#‘(◡𝑔 “ {0})) = (deg‘𝑔)) |
138 | | simp1l 1085 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → 𝑑 ∈ ℕ) |
139 | | simp3l 1089 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → (𝑑 + 1) = (deg‘𝑔)) |
140 | | simp2 1062 |
. . . . . . . . . 10
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
141 | 140, 132 | sylib 208 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
142 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑔 quot (Xp
∘𝑓 − (ℂ × {𝑧}))) = (𝑔 quot (Xp
∘𝑓 − (ℂ × {𝑧}))) |
143 | 133, 134,
135, 136, 137, 138, 139, 141, 142 | vieta1lem2 24066 |
. . . . . . . 8
⊢ (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
∧ ∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔))) → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) |
144 | 143 | 3exp 1264 |
. . . . . . 7
⊢ ((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
→ (∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (◡𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → (((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))))) |
145 | 132, 144 | syl5bir 233 |
. . . . . 6
⊢ ((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ))
→ (∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → (((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))))) |
146 | 145 | ralrimdva 2969 |
. . . . 5
⊢ (𝑑 ∈ ℕ →
(∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ∀𝑔 ∈
(Poly‘ℂ)(((𝑑 +
1) = (deg‘𝑔) ∧
(#‘(◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))))) |
147 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓)) |
148 | 147 | eqeq2d 2632 |
. . . . . . . 8
⊢ (𝑔 = 𝑓 → ((𝑑 + 1) = (deg‘𝑔) ↔ (𝑑 + 1) = (deg‘𝑓))) |
149 | | cnveq 5296 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝑓 → ◡𝑔 = ◡𝑓) |
150 | 149 | imaeq1d 5465 |
. . . . . . . . . 10
⊢ (𝑔 = 𝑓 → (◡𝑔 “ {0}) = (◡𝑓 “ {0})) |
151 | 150 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑔 = 𝑓 → (#‘(◡𝑔 “ {0})) = (#‘(◡𝑓 “ {0}))) |
152 | 151, 147 | eqeq12d 2637 |
. . . . . . . 8
⊢ (𝑔 = 𝑓 → ((#‘(◡𝑔 “ {0})) = (deg‘𝑔) ↔ (#‘(◡𝑓 “ {0})) = (deg‘𝑓))) |
153 | 148, 152 | anbi12d 747 |
. . . . . . 7
⊢ (𝑔 = 𝑓 → (((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)))) |
154 | 150 | sumeq1d 14431 |
. . . . . . . 8
⊢ (𝑔 = 𝑓 → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = Σ𝑥 ∈ (◡𝑓 “ {0})𝑥) |
155 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝑓 → (coeff‘𝑔) = (coeff‘𝑓)) |
156 | 147 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝑓 → ((deg‘𝑔) − 1) = ((deg‘𝑓) − 1)) |
157 | 155, 156 | fveq12d 6197 |
. . . . . . . . . 10
⊢ (𝑔 = 𝑓 → ((coeff‘𝑔)‘((deg‘𝑔) − 1)) = ((coeff‘𝑓)‘((deg‘𝑓) − 1))) |
158 | 155, 147 | fveq12d 6197 |
. . . . . . . . . 10
⊢ (𝑔 = 𝑓 → ((coeff‘𝑔)‘(deg‘𝑔)) = ((coeff‘𝑓)‘(deg‘𝑓))) |
159 | 157, 158 | oveq12d 6668 |
. . . . . . . . 9
⊢ (𝑔 = 𝑓 → (((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) /
((coeff‘𝑓)‘(deg‘𝑓)))) |
160 | 159 | negeqd 10275 |
. . . . . . . 8
⊢ (𝑔 = 𝑓 → -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) /
((coeff‘𝑓)‘(deg‘𝑓)))) |
161 | 154, 160 | eqeq12d 2637 |
. . . . . . 7
⊢ (𝑔 = 𝑓 → (Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) ↔ Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
162 | 153, 161 | imbi12d 334 |
. . . . . 6
⊢ (𝑔 = 𝑓 → ((((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) ↔ (((𝑑 + 1) = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
163 | 162 | cbvralv 3171 |
. . . . 5
⊢
(∀𝑔 ∈
(Poly‘ℂ)(((𝑑 +
1) = (deg‘𝑔) ∧
(#‘(◡𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (◡𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) ↔ ∀𝑓 ∈
(Poly‘ℂ)(((𝑑 +
1) = (deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
164 | 146, 163 | syl6ib 241 |
. . . 4
⊢ (𝑑 ∈ ℕ →
(∀𝑓 ∈
(Poly‘ℂ)((𝑑 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ∀𝑓 ∈
(Poly‘ℂ)(((𝑑 +
1) = (deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))) |
165 | 8, 12, 16, 20, 127, 164 | nnind 11038 |
. . 3
⊢ (𝑁 ∈ ℕ →
∀𝑓 ∈
(Poly‘ℂ)((𝑁 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
166 | 4, 165 | syl 17 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))) |
167 | | vieta1.5 |
. 2
⊢ (𝜑 → (#‘𝑅) = 𝑁) |
168 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹)) |
169 | 168 | eqeq2d 2632 |
. . . . . 6
⊢ (𝑓 = 𝐹 → (𝑁 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝐹))) |
170 | | cnveq 5296 |
. . . . . . . . . 10
⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) |
171 | 170 | imaeq1d 5465 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = (◡𝐹 “ {0})) |
172 | | vieta1.3 |
. . . . . . . . 9
⊢ 𝑅 = (◡𝐹 “ {0}) |
173 | 171, 172 | syl6eqr 2674 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (◡𝑓 “ {0}) = 𝑅) |
174 | 173 | fveq2d 6195 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → (#‘(◡𝑓 “ {0})) = (#‘𝑅)) |
175 | | vieta1.2 |
. . . . . . . 8
⊢ 𝑁 = (deg‘𝐹) |
176 | 168, 175 | syl6eqr 2674 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → (deg‘𝑓) = 𝑁) |
177 | 174, 176 | eqeq12d 2637 |
. . . . . 6
⊢ (𝑓 = 𝐹 → ((#‘(◡𝑓 “ {0})) = (deg‘𝑓) ↔ (#‘𝑅) = 𝑁)) |
178 | 169, 177 | anbi12d 747 |
. . . . 5
⊢ (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝐹) ∧ (#‘𝑅) = 𝑁))) |
179 | 175 | biantrur 527 |
. . . . 5
⊢
((#‘𝑅) = 𝑁 ↔ (𝑁 = (deg‘𝐹) ∧ (#‘𝑅) = 𝑁)) |
180 | 178, 179 | syl6bbr 278 |
. . . 4
⊢ (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) ↔ (#‘𝑅) = 𝑁)) |
181 | 173 | sumeq1d 14431 |
. . . . 5
⊢ (𝑓 = 𝐹 → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = Σ𝑥 ∈ 𝑅 𝑥) |
182 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹)) |
183 | | vieta1.1 |
. . . . . . . . 9
⊢ 𝐴 = (coeff‘𝐹) |
184 | 182, 183 | syl6eqr 2674 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴) |
185 | 176 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝑓 = 𝐹 → ((deg‘𝑓) − 1) = (𝑁 − 1)) |
186 | 184, 185 | fveq12d 6197 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = (𝐴‘(𝑁 − 1))) |
187 | 184, 176 | fveq12d 6197 |
. . . . . . 7
⊢ (𝑓 = 𝐹 → ((coeff‘𝑓)‘(deg‘𝑓)) = (𝐴‘𝑁)) |
188 | 186, 187 | oveq12d 6668 |
. . . . . 6
⊢ (𝑓 = 𝐹 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = ((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))) |
189 | 188 | negeqd 10275 |
. . . . 5
⊢ (𝑓 = 𝐹 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))) |
190 | 181, 189 | eqeq12d 2637 |
. . . 4
⊢ (𝑓 = 𝐹 → (Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁)))) |
191 | 180, 190 | imbi12d 334 |
. . 3
⊢ (𝑓 = 𝐹 → (((𝑁 = (deg‘𝑓) ∧ (#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((#‘𝑅) = 𝑁 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))))) |
192 | 191 | rspcv 3305 |
. 2
⊢ (𝐹 ∈ (Poly‘ℂ)
→ (∀𝑓 ∈
(Poly‘ℂ)((𝑁 =
(deg‘𝑓) ∧
(#‘(◡𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (◡𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ((#‘𝑅) = 𝑁 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))))) |
193 | 3, 166, 167, 192 | syl3c 66 |
1
⊢ (𝜑 → Σ𝑥 ∈ 𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴‘𝑁))) |