Proof of Theorem aannenlem1
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | breq2 5146 | . . . . . . 7
⊢ (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴)) | 
| 2 |  | breq2 5146 | . . . . . . . 8
⊢ (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)) | 
| 3 | 2 | ralbidv 3177 | . . . . . . 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 3443 | . . . . 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 3326 | . . . 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 3443 | . . 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 11237 | . . . 4
⊢ ℂ
∈ V | 
| 10 | 9 | rabex 5338 | . . 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 7015 | . 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 5051 | . . 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 14014 | . . . . . . 7
⊢ (-𝐴...𝐴) ∈ Fin | 
| 14 |  | fzfi 14014 | . . . . . . 7
⊢
(0...𝐴) ∈
Fin | 
| 15 |  | mapfi 9389 | . . . . . . 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 7465 | . . . . . 6
⊢ ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V | 
| 19 |  | neeq1 3002 | . . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝 ↔ 𝑎 ≠
0𝑝)) | 
| 20 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎)) | 
| 21 | 20 | breq1d 5152 | . . . . . . . . . . 11
⊢ (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴)) | 
| 22 |  | fveq2 6905 | . . . . . . . . . . . . . . 15
⊢ (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎)) | 
| 23 | 22 | fveq1d 6907 | . . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒)) | 
| 24 | 23 | fveq2d 6909 | . . . . . . . . . . . . 13
⊢ (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒))) | 
| 25 | 24 | breq1d 5152 | . . . . . . . . . . . 12
⊢ (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) | 
| 26 | 25 | ralbidv 3177 | . . . . . . . . . . 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 3691 | . . . . . . . . 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 1138 | . . . . . . . . . 10
⊢ ((𝑎 ≠ 0𝑝
∧ (deg‘𝑎) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) | 
| 30 | 29 | anim2i 617 | . . . . . . . . 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 217 | . . . . . . . 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 12626 | . . . . . . . . . . . . . . 15
⊢ 0 ∈
ℤ | 
| 33 |  | eqid 2736 | . . . . . . . . . . . . . . . 16
⊢
(coeff‘𝑎) =
(coeff‘𝑎) | 
| 34 | 33 | coef2 26271 | . . . . . . . . . . . . . . 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 6736 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎) Fn ℕ0) | 
| 38 | 35 | adantl 481 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) → (coeff‘𝑎):ℕ0⟶ℤ) | 
| 39 | 38 | ffvelcdmda 7103 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((coeff‘𝑎)‘𝑒) ∈ ℤ) | 
| 40 | 39 | zred 12724 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((coeff‘𝑎)‘𝑒) ∈ ℝ) | 
| 41 |  | nn0re 12537 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℝ) | 
| 42 | 41 | ad2antrr 726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈
ℝ) | 
| 43 | 40, 42 | absled 15470 | . . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴))) | 
| 44 |  | nn0z 12640 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℤ) | 
| 45 | 44 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈
ℤ) | 
| 46 | 45 | znegcld 12726 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈
ℤ) | 
| 47 |  | elfz 13554 | . . . . . . . . . . . . . . . . . 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 282 | . . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))) | 
| 50 | 49 | biimpd 229 | . . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) →
((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))) | 
| 51 | 50 | ralimdva 3166 | . . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℕ0
∧ 𝑎 ∈
(Poly‘ℤ)) → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0
((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))) | 
| 52 | 51 | impr 454 | . . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0
((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) | 
| 53 |  | fnfvrnss 7140 | . . . . . . . . . . . . 13
⊢
(((coeff‘𝑎) Fn
ℕ0 ∧ ∀𝑒 ∈ ℕ0
((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)) | 
| 54 | 37, 52, 53 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)) | 
| 55 |  | df-f 6564 | . . . . . . . . . . . 12
⊢
((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran
(coeff‘𝑎) ⊆
(-𝐴...𝐴))) | 
| 56 | 37, 54, 55 | sylanbrc 583 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶(-𝐴...𝐴)) | 
| 57 |  | fz0ssnn0 13663 | . . . . . . . . . . 11
⊢
(0...𝐴) ⊆
ℕ0 | 
| 58 |  | fssres 6773 | . . . . . . . . . . 11
⊢
(((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) →
((coeff‘𝑎) ↾
(0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴)) | 
| 59 | 56, 57, 58 | sylancl 586 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ (𝑎 ∈
(Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴)) | 
| 60 |  | ovex 7465 | . . . . . . . . . . 11
⊢ (-𝐴...𝐴) ∈ V | 
| 61 |  | ovex 7465 | . . . . . . . . . . 11
⊢
(0...𝐴) ∈
V | 
| 62 | 60, 61 | elmap 8912 | . . . . . . . . . 10
⊢
(((coeff‘𝑎)
↾ (0...𝐴)) ∈
((-𝐴...𝐴) ↑m (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴)) | 
| 63 | 59, 62 | sylibr 234 | . . . . . . . . 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 412 | . . . . . . . 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 1137 | . . . . . . . . . 10
⊢ ((𝑎 ≠ 0𝑝
∧ (deg‘𝑎) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴) | 
| 67 | 66 | anim2i 617 | . . . . . . . . 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 217 | . . . . . . . 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 3002 | . . . . . . . . . . 11
⊢ (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝 ↔ 𝑏 ≠
0𝑝)) | 
| 70 |  | fveq2 6905 | . . . . . . . . . . . 12
⊢ (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏)) | 
| 71 | 70 | breq1d 5152 | . . . . . . . . . . 11
⊢ (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴)) | 
| 72 |  | fveq2 6905 | . . . . . . . . . . . . . . 15
⊢ (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏)) | 
| 73 | 72 | fveq1d 6907 | . . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒)) | 
| 74 | 73 | fveq2d 6909 | . . . . . . . . . . . . 13
⊢ (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒))) | 
| 75 | 74 | breq1d 5152 | . . . . . . . . . . . 12
⊢ (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)) | 
| 76 | 75 | ralbidv 3177 | . . . . . . . . . . 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 3691 | . . . . . . . . 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 1137 | . . . . . . . . . 10
⊢ ((𝑏 ≠ 0𝑝
∧ (deg‘𝑏) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴) | 
| 80 | 79 | anim2i 617 | . . . . . . . . 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 217 | . . . . . . . 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 774 | . . . . . . . . . . . . 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 26238 | . . . . . . . . . . . . 13
⊢ (𝑎 ∈ (Poly‘ℤ)
→ 𝑎:ℂ⟶ℂ) | 
| 84 |  | ffn 6735 | . . . . . . . . . . . . 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 776 | . . . . . . . . . . . . 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 26238 | . . . . . . . . . . . . 13
⊢ (𝑏 ∈ (Poly‘ℤ)
→ 𝑏:ℂ⟶ℂ) | 
| 88 |  | ffn 6735 | . . . . . . . . . . . . 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 777 | . . . . . . . . . . . . . . . . . 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 480 | . . . . . . . . . . . . . . . . 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 6907 | . . . . . . . . . . . . . . . 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 6924 | . . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑)) | 
| 94 | 93 | adantl 481 | . . . . . . . . . . . . . . . 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 6924 | . . . . . . . . . . . . . . . . 17
⊢ (𝑑 ∈ (0...𝐴) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑)) | 
| 96 | 95 | adantl 481 | . . . . . . . . . . . . . . . 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 2784 | . . . . . . . . . . . . . . 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 7447 | . . . . . . . . . . . . . 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 15739 | . . . . . . . . . . . . 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 782 | . . . . . . . . . . . . . 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 783 | . . . . . . . . . . . . . . 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 26273 | . . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ (Poly‘ℤ)
→ (deg‘𝑎) ∈
ℕ0) | 
| 103 |  | nn0z 12640 | . . . . . . . . . . . . . . . . 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 776 | . . . . . . . . . . . . . . . . 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 12641 | . . . . . . . . . . . . . . . 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 12893 | . . . . . . . . . . . . . . . 16
⊢
(((deg‘𝑎)
∈ ℤ ∧ 𝐴
∈ ℤ) → (𝐴
∈ (ℤ≥‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴)) | 
| 108 | 104, 106,
107 | syl2anc 584 | . . . . . . . . . . . . . . 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 257 | . . . . . . . . . . . . . 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 484 | . . . . . . . . . . . . . 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 2736 | . . . . . . . . . . . . . . 15
⊢
(deg‘𝑎) =
(deg‘𝑎) | 
| 112 | 33, 111 | coeid3 26280 | . . . . . . . . . . . . . 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 1238 | . . . . . . . . . . . . . . 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 1118 | . . . . . . . . . . . . . 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 777 | . . . . . . . . . . . . . . . 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 480 | . . . . . . . . . . . . . . 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 26273 | . . . . . . . . . . . . . . . . 17
⊢ (𝑏 ∈ (Poly‘ℤ)
→ (deg‘𝑏) ∈
ℕ0) | 
| 119 |  | nn0z 12640 | . . . . . . . . . . . . . . . . 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 12893 | . . . . . . . . . . . . . . . 16
⊢
(((deg‘𝑏)
∈ ℤ ∧ 𝐴
∈ ℤ) → (𝐴
∈ (ℤ≥‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴)) | 
| 122 | 120, 106,
121 | syl2anc 584 | . . . . . . . . . . . . . . 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 257 | . . . . . . . . . . . . . 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 2736 | . . . . . . . . . . . . . . 15
⊢
(coeff‘𝑏) =
(coeff‘𝑏) | 
| 125 |  | eqid 2736 | . . . . . . . . . . . . . . 15
⊢
(deg‘𝑏) =
(deg‘𝑏) | 
| 126 | 124, 125 | coeid3 26280 | . . . . . . . . . . . . . 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 2786 | . . . . . . . . . . . 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 7053 | . . . . . . . . . . 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 456 | . . . . . . . . . 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 6905 | . . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏)) | 
| 132 | 131 | reseq1d 5995 | . . . . . . . . . 10
⊢ (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴))) | 
| 133 | 130, 132 | impbid1 225 | . . . . . . . . 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 413 | . . . . . . . 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 607 | . . . . . . 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 9034 | . . . . . 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 9230 | . . . . 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 584 | . . . 4
⊢ (𝐴 ∈ ℕ0
→ {𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin) | 
| 140 |  | neeq1 3002 | . . . . . . . . 9
⊢ (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝 ↔ 𝑐 ≠
0𝑝)) | 
| 141 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐)) | 
| 142 | 141 | breq1d 5152 | . . . . . . . . 9
⊢ (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴)) | 
| 143 |  | fveq2 6905 | . . . . . . . . . . . . 13
⊢ (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐)) | 
| 144 | 143 | fveq1d 6907 | . . . . . . . . . . . 12
⊢ (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒)) | 
| 145 | 144 | fveq2d 6909 | . . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒))) | 
| 146 | 145 | breq1d 5152 | . . . . . . . . . 10
⊢ (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)) | 
| 147 | 146 | ralbidv 3177 | . . . . . . . . 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 3691 | . . . . . . 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 1136 | . . . . . . . 8
⊢ ((𝑐 ≠ 0𝑝
∧ (deg‘𝑐) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝) | 
| 151 | 150 | anim2i 617 | . . . . . . 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 217 | . . . . . 6
⊢ (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝐴 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠
0𝑝)) | 
| 153 |  | fveqeq2 6914 | . . . . . . . . . . 11
⊢ (𝑏 = 𝑎 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑎) = 0)) | 
| 154 | 153 | elrab 3691 | . . . . . . . . . 10
⊢ (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐‘𝑎) = 0)) | 
| 155 |  | plyf 26238 | . . . . . . . . . . . . 13
⊢ (𝑐 ∈ (Poly‘ℤ)
→ 𝑐:ℂ⟶ℂ) | 
| 156 | 155 | ffnd 6736 | . . . . . . . . . . . 12
⊢ (𝑐 ∈ (Poly‘ℤ)
→ 𝑐 Fn
ℂ) | 
| 157 | 156 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → 𝑐 Fn ℂ) | 
| 158 |  | fniniseg 7079 | . . . . . . . . . . 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 290 | . . . . . . . . 9
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} ↔ 𝑎 ∈ (◡𝑐 “ {0}))) | 
| 161 | 160 | eqrdv 2734 | . . . . . . . 8
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐‘𝑏) = 0} = (◡𝑐 “ {0})) | 
| 162 |  | eqid 2736 | . . . . . . . . . 10
⊢ (◡𝑐 “ {0}) = (◡𝑐 “ {0}) | 
| 163 | 162 | fta1 26351 | . . . . . . . . 9
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → ((◡𝑐 “ {0}) ∈ Fin ∧
(♯‘(◡𝑐 “ {0})) ≤ (deg‘𝑐))) | 
| 164 | 163 | simpld 494 | . . . . . . . 8
⊢ ((𝑐 ∈ (Poly‘ℤ)
∧ 𝑐 ≠
0𝑝) → (◡𝑐 “ {0}) ∈
Fin) | 
| 165 | 161, 164 | eqeltrd 2840 | . . . . . . 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 3144 | . . . 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 9384 | . . . 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 584 | . . 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 2845 | . 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 2840 | 1
⊢ (𝐴 ∈ ℕ0
→ (𝐻‘𝐴) ∈ Fin) |