Proof of Theorem aannenlem1
Step | Hyp | Ref
| Expression |
1 | | breq2 5035 |
. . . . . . 7
⊢ (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴)) |
2 | | breq2 5035 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)) |
3 | 2 | ralbidv 3110 |
. . . . . . 7
⊢ (𝑎 = 𝐴 → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)) |
4 | 1, 3 | 3anbi23d 1440 |
. . . . . 6
⊢ (𝑎 = 𝐴 → ((𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))) |
5 | 4 | rabbidv 3382 |
. . . . 5
⊢ (𝑎 = 𝐴 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} = {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)}) |
6 | 5 | rexeqdv 3318 |
. . . 4
⊢ (𝑎 = 𝐴 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0)) |
7 | 6 | rabbidv 3382 |
. . 3
⊢ (𝑎 = 𝐴 → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0}) |
8 | | aannenlem.a |
. . 3
⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) |
9 | | cnex 10699 |
. . . 4
⊢ ℂ
∈ V |
10 | 9 | rabex 5201 |
. . 3
⊢ {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0} ∈ V |
11 | 7, 8, 10 | fvmpt 6778 |
. 2
⊢ (𝐴 ∈ ℕ0
→ (𝐻‘𝐴) = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0}) |
12 | | iunrab 4939 |
. . 3
⊢ ∪ 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0} |
13 | | fzfi 13434 |
. . . . . . 7
⊢ (-𝐴...𝐴) ∈ Fin |
14 | | fzfi 13434 |
. . . . . . 7
⊢
(0...𝐴) ∈
Fin |
15 | | mapfi 8896 |
. . . . . . 7
⊢ (((-𝐴...𝐴) ∈ Fin ∧ (0...𝐴) ∈ Fin) → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin) |
16 | 13, 14, 15 | mp2an 692 |
. . . . . 6
⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin |
17 | 16 | a1i 11 |
. . . . 5
⊢ (𝐴 ∈ ℕ0
→ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin) |
18 | | ovex 7206 |
. . . . . 6
⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V |
19 | | neeq1 2997 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝 ↔ 𝑎 ≠
0𝑝)) |
20 | | fveq2 6677 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎)) |
21 | 20 | breq1d 5041 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴)) |
22 | | fveq2 6677 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎)) |
23 | 22 | fveq1d 6679 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒)) |
24 | 23 | fveq2d 6681 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒))) |
25 | 24 | breq1d 5041 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) |
26 | 25 | ralbidv 3110 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) |
27 | 19, 21, 26 | 3anbi123d 1437 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑎 → ((𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑎 ≠ 0𝑝 ∧
(deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))) |
28 | 27 | elrab 3589 |
. . . . . . . . 9
⊢ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝
∧ (deg‘𝑎) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))) |
29 | | simp3 1139 |
. . . . . . . . . 10
⊢ ((𝑎 ≠ 0𝑝
∧ (deg‘𝑎) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) |
30 | 29 | anim2i 620 |
. . . . . . . . 9
⊢ ((𝑎 ∈ (Poly‘ℤ)
∧ (𝑎 ≠
0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (𝑎 ∈ (Poly‘ℤ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) |
31 | 28, 30 | sylbi 220 |
. . . . . . . 8
⊢ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑎 ∈ (Poly‘ℤ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) |
32 | | 0z 12076 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
ℤ |
33 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢
(coeff‘𝑎) =
(coeff‘𝑎) |
34 | 33 | coef2 24983 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ (Poly‘ℤ)
∧ 0 ∈ ℤ) → (coeff‘𝑎):ℕ0⟶ℤ) |
35 | 32, 34 | mpan2 691 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ (Poly‘ℤ)
→ (coeff‘𝑎):ℕ0⟶ℤ) |
36 | 35 | ad2antrl 728 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶ℤ) |
37 | 36 | ffnd 6506 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎) Fn ℕ0) |
38 | 35 | adantl 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) → (coeff‘𝑎):ℕ0⟶ℤ) |
39 | 38 | ffvelrnda 6864 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((coeff‘𝑎)‘𝑒) ∈ ℤ) |
40 | 39 | zred 12171 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((coeff‘𝑎)‘𝑒) ∈ ℝ) |
41 | | nn0re 11988 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℝ) |
42 | 41 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈
ℝ) |
43 | 40, 42 | absled 14883 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴))) |
44 | | nn0z 12089 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℤ) |
45 | 44 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈
ℤ) |
46 | 45 | znegcld 12173 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈
ℤ) |
47 | | elfz 12990 |
. . . . . . . . . . . . . . . . . 18
⊢
((((coeff‘𝑎)‘𝑒) ∈ ℤ ∧ -𝐴 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴) ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴))) |
48 | 39, 46, 45, 47 | syl3anc 1372 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
(((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴) ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴))) |
49 | 43, 48 | bitr4d 285 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))) |
50 | 49 | biimpd 232 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))) |
51 | 50 | ralimdva 3092 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0
((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))) |
52 | 51 | impr 458 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0
((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) |
53 | | fnfvrnss 6897 |
. . . . . . . . . . . . 13
⊢
(((coeff‘𝑎) Fn
ℕ0 ∧ ∀𝑒 ∈ ℕ0
((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)) |
54 | 37, 52, 53 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)) |
55 | | df-f 6344 |
. . . . . . . . . . . 12
⊢
((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran
(coeff‘𝑎) ⊆
(-𝐴...𝐴))) |
56 | 37, 54, 55 | sylanbrc 586 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶(-𝐴...𝐴)) |
57 | | fz0ssnn0 13096 |
. . . . . . . . . . 11
⊢
(0...𝐴) ⊆
ℕ0 |
58 | | fssres 6545 |
. . . . . . . . . . 11
⊢
(((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) →
((coeff‘𝑎) ↾
(0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴)) |
59 | 56, 57, 58 | sylancl 589 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴)) |
60 | | ovex 7206 |
. . . . . . . . . . 11
⊢ (-𝐴...𝐴) ∈ V |
61 | | ovex 7206 |
. . . . . . . . . . 11
⊢
(0...𝐴) ∈
V |
62 | 60, 61 | elmap 8484 |
. . . . . . . . . 10
⊢
(((coeff‘𝑎)
↾ (0...𝐴)) ∈
((-𝐴...𝐴) ↑m (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴)) |
63 | 59, 62 | sylibr 237 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴))) |
64 | 63 | ex 416 |
. . . . . . . 8
⊢ (𝐴 ∈ ℕ0
→ ((𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)))) |
65 | 31, 64 | syl5 34 |
. . . . . . 7
⊢ (𝐴 ∈ ℕ0
→ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)))) |
66 | | simp2 1138 |
. . . . . . . . . 10
⊢ ((𝑎 ≠ 0𝑝
∧ (deg‘𝑎) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴) |
67 | 66 | anim2i 620 |
. . . . . . . . 9
⊢ ((𝑎 ∈ (Poly‘ℤ)
∧ (𝑎 ≠
0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (𝑎 ∈ (Poly‘ℤ) ∧
(deg‘𝑎) ≤ 𝐴)) |
68 | 28, 67 | sylbi 220 |
. . . . . . . 8
⊢ (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑎 ∈ (Poly‘ℤ) ∧
(deg‘𝑎) ≤ 𝐴)) |
69 | | neeq1 2997 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝 ↔ 𝑏 ≠
0𝑝)) |
70 | | fveq2 6677 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏)) |
71 | 70 | breq1d 5041 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴)) |
72 | | fveq2 6677 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏)) |
73 | 72 | fveq1d 6679 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒)) |
74 | 73 | fveq2d 6681 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒))) |
75 | 74 | breq1d 5041 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)) |
76 | 75 | ralbidv 3110 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑏 → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)) |
77 | 69, 71, 76 | 3anbi123d 1437 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑏 → ((𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑏 ≠ 0𝑝 ∧
(deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))) |
78 | 77 | elrab 3589 |
. . . . . . . . 9
⊢ (𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑏 ∈ (Poly‘ℤ) ∧ (𝑏 ≠ 0𝑝
∧ (deg‘𝑏) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))) |
79 | | simp2 1138 |
. . . . . . . . . 10
⊢ ((𝑏 ≠ 0𝑝
∧ (deg‘𝑏) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴) |
80 | 79 | anim2i 620 |
. . . . . . . . 9
⊢ ((𝑏 ∈ (Poly‘ℤ)
∧ (𝑏 ≠
0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)) → (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) |
81 | 78, 80 | sylbi 220 |
. . . . . . . 8
⊢ (𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) |
82 | | simplll 775 |
. . . . . . . . . . . . 13
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) → 𝑎 ∈
(Poly‘ℤ)) |
83 | | plyf 24950 |
. . . . . . . . . . . . 13
⊢ (𝑎 ∈ (Poly‘ℤ)
→ 𝑎:ℂ⟶ℂ) |
84 | | ffn 6505 |
. . . . . . . . . . . . 13
⊢ (𝑎:ℂ⟶ℂ →
𝑎 Fn
ℂ) |
85 | 82, 83, 84 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) → 𝑎 Fn ℂ) |
86 | | simplrl 777 |
. . . . . . . . . . . . 13
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) → 𝑏 ∈
(Poly‘ℤ)) |
87 | | plyf 24950 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ (Poly‘ℤ)
→ 𝑏:ℂ⟶ℂ) |
88 | | ffn 6505 |
. . . . . . . . . . . . 13
⊢ (𝑏:ℂ⟶ℂ →
𝑏 Fn
ℂ) |
89 | 86, 87, 88 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) → 𝑏 Fn ℂ) |
90 | | simplrr 778 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) →
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴))) |
91 | 90 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴))) |
92 | 91 | fveq1d 6679 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑)) |
93 | | fvres 6696 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑)) |
94 | 93 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑)) |
95 | | fvres 6696 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑)) |
96 | 95 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢
((((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑)) |
97 | 92, 94, 96 | 3eqtr3d 2782 |
. . . . . . . . . . . . . . 15
⊢
((((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → ((coeff‘𝑎)‘𝑑) = ((coeff‘𝑏)‘𝑑)) |
98 | 97 | oveq1d 7188 |
. . . . . . . . . . . . . 14
⊢
((((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑)) = (((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑))) |
99 | 98 | sumeq2dv 15156 |
. . . . . . . . . . . . 13
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) →
Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑)) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑))) |
100 | | simp-4l 783 |
. . . . . . . . . . . . . 14
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑎 ∈
(Poly‘ℤ)) |
101 | | simp-4r 784 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) →
(deg‘𝑎) ≤ 𝐴) |
102 | | dgrcl 24985 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ (Poly‘ℤ)
→ (deg‘𝑎) ∈
ℕ0) |
103 | | nn0z 12089 |
. . . . . . . . . . . . . . . . 17
⊢
((deg‘𝑎)
∈ ℕ0 → (deg‘𝑎) ∈ ℤ) |
104 | 100, 102,
103 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) →
(deg‘𝑎) ∈
ℤ) |
105 | | simplrl 777 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈
ℕ0) |
106 | 105 | nn0zd 12169 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈
ℤ) |
107 | | eluz 12341 |
. . . . . . . . . . . . . . . 16
⊢
(((deg‘𝑎)
∈ ℤ ∧ 𝐴
∈ ℤ) → (𝐴
∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴)) |
108 | 104, 106,
107 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝐴 ∈
(ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴)) |
109 | 101, 108 | mpbird 260 |
. . . . . . . . . . . . . 14
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈
(ℤ≥‘(deg‘𝑎))) |
110 | | simpr 488 |
. . . . . . . . . . . . . 14
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑐 ∈
ℂ) |
111 | | eqid 2739 |
. . . . . . . . . . . . . . 15
⊢
(deg‘𝑎) =
(deg‘𝑎) |
112 | 33, 111 | coeid3 24992 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ (Poly‘ℤ)
∧ 𝐴 ∈
(ℤ≥‘(deg‘𝑎)) ∧ 𝑐 ∈ ℂ) → (𝑎‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑))) |
113 | 100, 109,
110, 112 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑎‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐↑𝑑))) |
114 | | simp1rl 1239 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴))) ∧ 𝑐 ∈ ℂ) → 𝑏 ∈
(Poly‘ℤ)) |
115 | 114 | 3expa 1119 |
. . . . . . . . . . . . . 14
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑏 ∈
(Poly‘ℤ)) |
116 | | simplrr 778 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) →
(deg‘𝑏) ≤ 𝐴) |
117 | 116 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) →
(deg‘𝑏) ≤ 𝐴) |
118 | | dgrcl 24985 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ∈ (Poly‘ℤ)
→ (deg‘𝑏) ∈
ℕ0) |
119 | | nn0z 12089 |
. . . . . . . . . . . . . . . . 17
⊢
((deg‘𝑏)
∈ ℕ0 → (deg‘𝑏) ∈ ℤ) |
120 | 115, 118,
119 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) →
(deg‘𝑏) ∈
ℤ) |
121 | | eluz 12341 |
. . . . . . . . . . . . . . . 16
⊢
(((deg‘𝑏)
∈ ℤ ∧ 𝐴
∈ ℤ) → (𝐴
∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴)) |
122 | 120, 106,
121 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝐴 ∈
(ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴)) |
123 | 117, 122 | mpbird 260 |
. . . . . . . . . . . . . 14
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈
(ℤ≥‘(deg‘𝑏))) |
124 | | eqid 2739 |
. . . . . . . . . . . . . . 15
⊢
(coeff‘𝑏) =
(coeff‘𝑏) |
125 | | eqid 2739 |
. . . . . . . . . . . . . . 15
⊢
(deg‘𝑏) =
(deg‘𝑏) |
126 | 124, 125 | coeid3 24992 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 ∈ (Poly‘ℤ)
∧ 𝐴 ∈
(ℤ≥‘(deg‘𝑏)) ∧ 𝑐 ∈ ℂ) → (𝑏‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑))) |
127 | 115, 123,
110, 126 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑏‘𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐↑𝑑))) |
128 | 99, 113, 127 | 3eqtr4d 2784 |
. . . . . . . . . . . 12
⊢
(((((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧
((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑎‘𝑐) = (𝑏‘𝑐)) |
129 | 85, 89, 128 | eqfnfvd 6815 |
. . . . . . . . . . 11
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ (𝐴 ∈ ℕ0
∧ ((coeff‘𝑎)
↾ (0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)))) → 𝑎 = 𝑏) |
130 | 129 | expr 460 |
. . . . . . . . . 10
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ 𝐴 ∈ ℕ0) →
(((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)) → 𝑎 = 𝑏)) |
131 | | fveq2 6677 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏)) |
132 | 131 | reseq1d 5825 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴))) |
133 | 130, 132 | impbid1 228 |
. . . . . . . . 9
⊢ ((((𝑎 ∈ (Poly‘ℤ)
∧ (deg‘𝑎) ≤
𝐴) ∧ (𝑏 ∈ (Poly‘ℤ)
∧ (deg‘𝑏) ≤
𝐴)) ∧ 𝐴 ∈ ℕ0) →
(((coeff‘𝑎) ↾
(0...𝐴)) =
((coeff‘𝑏) ↾
(0...𝐴)) ↔ 𝑎 = 𝑏)) |
134 | 133 | expcom 417 |
. . . . . . . 8
⊢ (𝐴 ∈ ℕ0
→ (((𝑎 ∈
(Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧
(deg‘𝑏) ≤ 𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏))) |
135 | 68, 81, 134 | syl2ani 610 |
. . . . . . 7
⊢ (𝐴 ∈ ℕ0
→ ((𝑎 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∧ 𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)}) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏))) |
136 | 65, 135 | dom2d 8599 |
. . . . . 6
⊢ (𝐴 ∈ ℕ0
→ (((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V → {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴)))) |
137 | 18, 136 | mpi 20 |
. . . . 5
⊢ (𝐴 ∈ ℕ0
→ {𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴))) |
138 | | domfi 8820 |
. . . . 5
⊢
((((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin ∧ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴))) → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin) |
139 | 17, 137, 138 | syl2anc 587 |
. . . 4
⊢ (𝐴 ∈ ℕ0
→ {𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin) |
140 | | neeq1 2997 |
. . . . . . . . 9
⊢ (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝 ↔ 𝑐 ≠
0𝑝)) |
141 | | fveq2 6677 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐)) |
142 | 141 | breq1d 5041 |
. . . . . . . . 9
⊢ (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴)) |
143 | | fveq2 6677 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐)) |
144 | 143 | fveq1d 6679 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒)) |
145 | 144 | fveq2d 6681 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒))) |
146 | 145 | breq1d 5041 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)) |
147 | 146 | ralbidv 3110 |
. . . . . . . . 9
⊢ (𝑑 = 𝑐 → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)) |
148 | 140, 142,
147 | 3anbi123d 1437 |
. . . . . . . 8
⊢ (𝑑 = 𝑐 → ((𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑐 ≠ 0𝑝 ∧
(deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))) |
149 | 148 | elrab 3589 |
. . . . . . 7
⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑐 ∈ (Poly‘ℤ) ∧ (𝑐 ≠ 0𝑝
∧ (deg‘𝑐) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))) |
150 | | simp1 1137 |
. . . . . . . 8
⊢ ((𝑐 ≠ 0𝑝
∧ (deg‘𝑐) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝) |
151 | 150 | anim2i 620 |
. . . . . . 7
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ (𝑐 ≠
0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)) → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠
0𝑝)) |
152 | 149, 151 | sylbi 220 |
. . . . . 6
⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠
0𝑝)) |
153 | | fveqeq2 6686 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑎 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑎) = 0)) |
154 | 153 | elrab 3589 |
. . . . . . . . . 10
⊢ (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)) |
155 | | plyf 24950 |
. . . . . . . . . . . . 13
⊢ (𝑐 ∈ (Poly‘ℤ)
→ 𝑐:ℂ⟶ℂ) |
156 | 155 | ffnd 6506 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ (Poly‘ℤ)
→ 𝑐 Fn
ℂ) |
157 | 156 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → 𝑐 Fn ℂ) |
158 | | fniniseg 6840 |
. . . . . . . . . . 11
⊢ (𝑐 Fn ℂ → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0))) |
159 | 157, 158 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → (𝑎 ∈ (◡𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0))) |
160 | 154, 159 | bitr4id 293 |
. . . . . . . . 9
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ 𝑎 ∈ (◡𝑐 “ {0}))) |
161 | 160 | eqrdv 2737 |
. . . . . . . 8
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = (◡𝑐 “ {0})) |
162 | | eqid 2739 |
. . . . . . . . . 10
⊢ (◡𝑐 “ {0}) = (◡𝑐 “ {0}) |
163 | 162 | fta1 25059 |
. . . . . . . . 9
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → ((◡𝑐 “ {0}) ∈ Fin ∧
(♯‘(◡𝑐 “ {0})) ≤ (deg‘𝑐))) |
164 | 163 | simpld 498 |
. . . . . . . 8
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → (◡𝑐 “ {0}) ∈
Fin) |
165 | 161, 164 | eqeltrd 2834 |
. . . . . . 7
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin) |
166 | 165 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ ℕ0
→ ((𝑐 ∈
(Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)) |
167 | 152, 166 | syl5 34 |
. . . . 5
⊢ (𝐴 ∈ ℕ0
→ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin)) |
168 | 167 | ralrimiv 3096 |
. . . 4
⊢ (𝐴 ∈ ℕ0
→ ∀𝑐 ∈
{𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin) |
169 | | iunfi 8888 |
. . . 4
⊢ (({𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin ∧ ∀𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin) → ∪ 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin) |
170 | 139, 168,
169 | syl2anc 587 |
. . 3
⊢ (𝐴 ∈ ℕ0
→ ∪ 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ∈ Fin) |
171 | 12, 170 | eqeltrrid 2839 |
. 2
⊢ (𝐴 ∈ ℕ0
→ {𝑏 ∈ ℂ
∣ ∃𝑐 ∈
{𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐‘𝑏) = 0} ∈ Fin) |
172 | 11, 171 | eqeltrd 2834 |
1
⊢ (𝐴 ∈ ℕ0
→ (𝐻‘𝐴) ∈ Fin) |