| Step | Hyp | Ref
| Expression |
| 1 | | fveqeq2 6915 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑔 → ((𝑐‘𝑏) = 0 ↔ (𝑐‘𝑔) = 0)) |
| 2 | 1 | rexbidv 3179 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑔 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑔) = 0)) |
| 3 | | simp3 1139 |
. . . . . . . . . 10
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ ℂ) |
| 4 | | neeq1 3003 |
. . . . . . . . . . . . 13
⊢ (𝑑 = ℎ → (𝑑 ≠ 0𝑝 ↔ ℎ ≠
0𝑝)) |
| 5 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = ℎ → (deg‘𝑑) = (deg‘ℎ)) |
| 6 | 5 | breq1d 5153 |
. . . . . . . . . . . . 13
⊢ (𝑑 = ℎ → ((deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ↔
(deg‘ℎ) ≤ sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 7 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 = ℎ → (coeff‘𝑑) = (coeff‘ℎ)) |
| 8 | 7 | fveq1d 6908 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 = ℎ → ((coeff‘𝑑)‘𝑒) = ((coeff‘ℎ)‘𝑒)) |
| 9 | 8 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 = ℎ → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒))) |
| 10 | 9 | breq1d 5153 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = ℎ → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ↔
(abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 11 | 10 | ralbidv 3178 |
. . . . . . . . . . . . 13
⊢ (𝑑 = ℎ → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ↔
∀𝑒 ∈
ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 12 | 4, 6, 11 | 3anbi123d 1438 |
. . . . . . . . . . . 12
⊢ (𝑑 = ℎ → ((𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < )) ↔
(ℎ ≠
0𝑝 ∧ (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)))) |
| 13 | | eldifi 4131 |
. . . . . . . . . . . . . 14
⊢ (ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) → ℎ ∈ (Poly‘ℤ)) |
| 14 | 13 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ)) |
| 15 | 14 | 3adant2 1132 |
. . . . . . . . . . . 12
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ℎ ∈ (Poly‘ℤ)) |
| 16 | | eldifsni 4790 |
. . . . . . . . . . . . . . 15
⊢ (ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) → ℎ ≠ 0𝑝) |
| 17 | 16 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ℎ ≠ 0𝑝) |
| 18 | | 0nn0 12541 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℕ0 |
| 19 | | dgrcl 26272 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ ∈ (Poly‘ℤ)
→ (deg‘ℎ) ∈
ℕ0) |
| 20 | 14, 19 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘ℎ) ∈
ℕ0) |
| 21 | | prssi 4821 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℕ0 ∧ (deg‘ℎ) ∈ ℕ0) → {0,
(deg‘ℎ)} ⊆
ℕ0) |
| 22 | 18, 20, 21 | sylancr 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {0, (deg‘ℎ)} ⊆
ℕ0) |
| 23 | | ssrab2 4080 |
. . . . . . . . . . . . . . . . . 18
⊢ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ⊆
ℕ0 |
| 24 | 23 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ⊆
ℕ0) |
| 25 | 22, 24 | unssd 4192 |
. . . . . . . . . . . . . . . 16
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆
ℕ0) |
| 26 | | nn0ssre 12530 |
. . . . . . . . . . . . . . . . 17
⊢
ℕ0 ⊆ ℝ |
| 27 | | ressxr 11305 |
. . . . . . . . . . . . . . . . 17
⊢ ℝ
⊆ ℝ* |
| 28 | 26, 27 | sstri 3993 |
. . . . . . . . . . . . . . . 16
⊢
ℕ0 ⊆ ℝ* |
| 29 | 25, 28 | sstrdi 3996 |
. . . . . . . . . . . . . . 15
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆
ℝ*) |
| 30 | | fvex 6919 |
. . . . . . . . . . . . . . . . 17
⊢
(deg‘ℎ) ∈
V |
| 31 | 30 | prid2 4763 |
. . . . . . . . . . . . . . . 16
⊢
(deg‘ℎ) ∈
{0, (deg‘ℎ)} |
| 32 | | elun1 4182 |
. . . . . . . . . . . . . . . 16
⊢
((deg‘ℎ) ∈
{0, (deg‘ℎ)} →
(deg‘ℎ) ∈ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 33 | 31, 32 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
(deg‘ℎ) ∈
({0, (deg‘ℎ)} ∪
{𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) |
| 34 | | supxrub 13366 |
. . . . . . . . . . . . . . 15
⊢ ((({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ* ∧
(deg‘ℎ) ∈ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) → (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)) |
| 35 | 29, 33, 34 | sylancl 586 |
. . . . . . . . . . . . . 14
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (deg‘ℎ) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)) |
| 36 | 29 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) → ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆
ℝ*) |
| 37 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘0)) |
| 38 | | abs0 15324 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(abs‘0) = 0 |
| 39 | 37, 38 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) = 0) |
| 40 | | c0ex 11255 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 0 ∈
V |
| 41 | 40 | prid1 4762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
{0, (deg‘ℎ)} |
| 42 | | elun1 4182 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 ∈
{0, (deg‘ℎ)} → 0
∈ ({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))})) |
| 43 | 41, 42 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
({0, (deg‘ℎ)} ∪
{𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) |
| 44 | 39, 43 | eqeltrdi 2849 |
. . . . . . . . . . . . . . . . . 18
⊢
(((coeff‘ℎ)‘𝑒) = 0 → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 45 | 44 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) = 0) →
(abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 46 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 =
(abs‘((coeff‘ℎ)‘𝑒)) → (𝑔 = (abs‘((coeff‘ℎ)‘𝑖)) ↔ (abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖)))) |
| 47 | 46 | rexbidv 3179 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 =
(abs‘((coeff‘ℎ)‘𝑒)) → (∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖)) ↔ ∃𝑖 ∈ (0...(deg‘ℎ))(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖)))) |
| 48 | | 0z 12624 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℤ |
| 49 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(coeff‘ℎ) =
(coeff‘ℎ) |
| 50 | 49 | coef2 26270 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((ℎ ∈ (Poly‘ℤ)
∧ 0 ∈ ℤ) → (coeff‘ℎ):ℕ0⟶ℤ) |
| 51 | 14, 48, 50 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (coeff‘ℎ):ℕ0⟶ℤ) |
| 52 | 51 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) →
((coeff‘ℎ)‘𝑒) ∈
ℤ) |
| 53 | | nn0abscl 15351 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((coeff‘ℎ)‘𝑒) ∈ ℤ →
(abs‘((coeff‘ℎ)‘𝑒)) ∈
ℕ0) |
| 54 | 52, 53 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) →
(abs‘((coeff‘ℎ)‘𝑒)) ∈
ℕ0) |
| 55 | 54 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) →
(abs‘((coeff‘ℎ)‘𝑒)) ∈
ℕ0) |
| 56 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ∈
ℕ0) |
| 57 | 20 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) →
(deg‘ℎ) ∈
ℕ0) |
| 58 | 14 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) → ℎ ∈
(Poly‘ℤ)) |
| 59 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) →
((coeff‘ℎ)‘𝑒) ≠ 0) |
| 60 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(deg‘ℎ) =
(deg‘ℎ) |
| 61 | 49, 60 | dgrub 26273 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ℎ ∈ (Poly‘ℤ)
∧ 𝑒 ∈
ℕ0 ∧ ((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ)) |
| 62 | 58, 56, 59, 61 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ≤ (deg‘ℎ)) |
| 63 | | elfz2nn0 13658 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑒 ∈ (0...(deg‘ℎ)) ↔ (𝑒 ∈ ℕ0 ∧
(deg‘ℎ) ∈
ℕ0 ∧ 𝑒
≤ (deg‘ℎ))) |
| 64 | 56, 57, 62, 63 | syl3anbrc 1344 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) → 𝑒 ∈ (0...(deg‘ℎ))) |
| 65 | | eqid 2737 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒)) |
| 66 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 𝑒 → (abs‘((coeff‘ℎ)‘𝑖)) = (abs‘((coeff‘ℎ)‘𝑒))) |
| 67 | 66 | rspceeqv 3645 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑒 ∈ (0...(deg‘ℎ)) ∧
(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑒))) → ∃𝑖 ∈ (0...(deg‘ℎ))(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖))) |
| 68 | 64, 65, 67 | sylancl 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) → ∃𝑖 ∈ (0...(deg‘ℎ))(abs‘((coeff‘ℎ)‘𝑒)) = (abs‘((coeff‘ℎ)‘𝑖))) |
| 69 | 47, 55, 68 | elrabd 3694 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) →
(abs‘((coeff‘ℎ)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) |
| 70 | | elun2 4183 |
. . . . . . . . . . . . . . . . . 18
⊢
((abs‘((coeff‘ℎ)‘𝑒)) ∈ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} → (abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 71 | 69, 70 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) ∧
((coeff‘ℎ)‘𝑒) ≠ 0) →
(abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 72 | 45, 71 | pm2.61dane 3029 |
. . . . . . . . . . . . . . . 16
⊢ (((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) →
(abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 73 | | supxrub 13366 |
. . . . . . . . . . . . . . . 16
⊢ ((({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ* ∧
(abs‘((coeff‘ℎ)‘𝑒)) ∈ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) → (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)) |
| 74 | 36, 72, 73 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) ∧ 𝑒 ∈ ℕ0) →
(abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)) |
| 75 | 74 | ralrimiva 3146 |
. . . . . . . . . . . . . 14
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → ∀𝑒 ∈ ℕ0
(abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)) |
| 76 | 17, 35, 75 | 3jca 1129 |
. . . . . . . . . . . . 13
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → (ℎ ≠ 0𝑝 ∧
(deg‘ℎ) ≤ sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 77 | 76 | 3adant2 1132 |
. . . . . . . . . . . 12
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (ℎ ≠ 0𝑝 ∧
(deg‘ℎ) ≤ sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘ℎ)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 78 | 12, 15, 77 | elrabd 3694 |
. . . . . . . . . . 11
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))}) |
| 79 | | simp2 1138 |
. . . . . . . . . . 11
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → (ℎ‘𝑔) = 0) |
| 80 | | fveq1 6905 |
. . . . . . . . . . . . 13
⊢ (𝑐 = ℎ → (𝑐‘𝑔) = (ℎ‘𝑔)) |
| 81 | 80 | eqeq1d 2739 |
. . . . . . . . . . . 12
⊢ (𝑐 = ℎ → ((𝑐‘𝑔) = 0 ↔ (ℎ‘𝑔) = 0)) |
| 82 | 81 | rspcev 3622 |
. . . . . . . . . . 11
⊢ ((ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} ∧
(ℎ‘𝑔) = 0) → ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑔) = 0) |
| 83 | 78, 79, 82 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑔) = 0) |
| 84 | 2, 3, 83 | elrabd 3694 |
. . . . . . . . 9
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0}) |
| 85 | | prfi 9363 |
. . . . . . . . . . . . . 14
⊢ {0,
(deg‘ℎ)} ∈
Fin |
| 86 | | fzfi 14013 |
. . . . . . . . . . . . . . . 16
⊢
(0...(deg‘ℎ))
∈ Fin |
| 87 | | abrexfi 9392 |
. . . . . . . . . . . . . . . 16
⊢
((0...(deg‘ℎ))
∈ Fin → {𝑔
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin) |
| 88 | 86, 87 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin |
| 89 | | rabssab 4085 |
. . . . . . . . . . . . . . 15
⊢ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} |
| 90 | | ssfi 9213 |
. . . . . . . . . . . . . . 15
⊢ (({𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin ∧ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ⊆ {𝑔 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}) → {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin) |
| 91 | 88, 89, 90 | mp2an 692 |
. . . . . . . . . . . . . 14
⊢ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin |
| 92 | | unfi 9211 |
. . . . . . . . . . . . . 14
⊢ (({0,
(deg‘ℎ)} ∈ Fin
∧ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))} ∈ Fin) → ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin) |
| 93 | 85, 91, 92 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin |
| 94 | 33 | ne0ii 4344 |
. . . . . . . . . . . . 13
⊢ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅ |
| 95 | | xrltso 13183 |
. . . . . . . . . . . . . 14
⊢ < Or
ℝ* |
| 96 | | fisupcl 9509 |
. . . . . . . . . . . . . 14
⊢ (( <
Or ℝ* ∧ (({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅ ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*)) →
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 97 | 95, 96 | mpan 690 |
. . . . . . . . . . . . 13
⊢ ((({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ∈ Fin ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ≠ ∅ ∧ ({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}) ⊆ ℝ*) →
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 98 | 93, 94, 29, 97 | mp3an12i 1467 |
. . . . . . . . . . . 12
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈ ({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))})) |
| 99 | 25, 98 | sseldd 3984 |
. . . . . . . . . . 11
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ 𝑔 ∈ ℂ) → sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈
ℕ0) |
| 100 | 99 | 3adant2 1132 |
. . . . . . . . . 10
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈
ℕ0) |
| 101 | | eqidd 2738 |
. . . . . . . . . 10
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0}) |
| 102 | | breq2 5147 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 103 | | breq2 5147 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 104 | 103 | ralbidv 3178 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
(∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))) |
| 105 | 102, 104 | 3anbi23d 1441 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
((𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ sup(({0,
(deg‘ℎ)} ∪ {𝑔 ∈ ℕ0
∣ ∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
)))) |
| 106 | 105 | rabbidv 3444 |
. . . . . . . . . . . . 13
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
{𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} = {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, <
))}) |
| 107 | 106 | rexeqdv 3327 |
. . . . . . . . . . . 12
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
(∃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) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0)) |
| 108 | 107 | rabbidv 3444 |
. . . . . . . . . . 11
⊢ (𝑎 = sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) →
{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) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0}) |
| 109 | 108 | rspceeqv 3645 |
. . . . . . . . . 10
⊢
((sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∈
ℕ0 ∧ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0}) → ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) |
| 110 | 100, 101,
109 | syl2anc 584 |
. . . . . . . . 9
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑎 ∈ ℕ0
{𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) |
| 111 | | cnex 11236 |
. . . . . . . . . . 11
⊢ ℂ
∈ V |
| 112 | 111 | rabex 5339 |
. . . . . . . . . 10
⊢ {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} ∈ V |
| 113 | | eleq2 2830 |
. . . . . . . . . . 11
⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 ↔ 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0})) |
| 114 | | eqeq1 2741 |
. . . . . . . . . . . 12
⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → (𝑓 = {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) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 115 | 114 | rexbidv 3179 |
. . . . . . . . . . 11
⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → (∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} ↔ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 116 | 113, 115 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} → ((𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) ↔ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} ∧ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))) |
| 117 | 112, 116 | spcev 3606 |
. . . . . . . . 9
⊢ ((𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
sup(({0, (deg‘ℎ)}
∪ {𝑔 ∈
ℕ0 ∣ ∃𝑖 ∈ (0...(deg‘ℎ))𝑔 = (abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} ∧ ∃𝑎 ∈ ℕ0 {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ) ∧
∀𝑒 ∈
ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ sup(({0, (deg‘ℎ)} ∪ {𝑔 ∈ ℕ0 ∣
∃𝑖 ∈
(0...(deg‘ℎ))𝑔 =
(abs‘((coeff‘ℎ)‘𝑖))}), ℝ*, < ))} (𝑐‘𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 118 | 84, 110, 117 | syl2anc 584 |
. . . . . . . 8
⊢ ((ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ∧ (ℎ‘𝑔) = 0 ∧ 𝑔 ∈ ℂ) → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 119 | 118 | 3exp 1120 |
. . . . . . 7
⊢ (ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) → ((ℎ‘𝑔) = 0 → (𝑔 ∈ ℂ → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})))) |
| 120 | 119 | rexlimiv 3148 |
. . . . . 6
⊢
(∃ℎ ∈
((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0 → (𝑔 ∈ ℂ → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}))) |
| 121 | 120 | impcom 407 |
. . . . 5
⊢ ((𝑔 ∈ ℂ ∧
∃ℎ ∈
((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0) → ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 122 | | eleq2 2830 |
. . . . . . . . 9
⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 ↔ 𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 123 | 1 | rexbidv 3179 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑔 → (∃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‘𝑔) = 0)) |
| 124 | 123 | elrab 3692 |
. . . . . . . . . 10
⊢ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} ↔ (𝑔 ∈ ℂ ∧ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0)) |
| 125 | | simp1 1137 |
. . . . . . . . . . . . . . 15
⊢ ((ℎ ≠ 0𝑝 ∧
(deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎) → ℎ ≠ 0𝑝) |
| 126 | 125 | anim2i 617 |
. . . . . . . . . . . . . 14
⊢ ((ℎ ∈ (Poly‘ℤ)
∧ (ℎ ≠
0𝑝 ∧ (deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)) → (ℎ ∈ (Poly‘ℤ) ∧ ℎ ≠
0𝑝)) |
| 127 | 5 | breq1d 5153 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 = ℎ → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘ℎ) ≤ 𝑎)) |
| 128 | 9 | breq1d 5153 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑑 = ℎ → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)) |
| 129 | 128 | ralbidv 3178 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 = ℎ → (∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎)) |
| 130 | 4, 127, 129 | 3anbi123d 1438 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 = ℎ → ((𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (ℎ ≠ 0𝑝 ∧
(deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))) |
| 131 | 130 | elrab 3692 |
. . . . . . . . . . . . . 14
⊢ (ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ↔ (ℎ ∈ (Poly‘ℤ) ∧ (ℎ ≠ 0𝑝 ∧
(deg‘ℎ) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘ℎ)‘𝑒)) ≤ 𝑎))) |
| 132 | | eldifsn 4786 |
. . . . . . . . . . . . . 14
⊢ (ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝}) ↔ (ℎ ∈ (Poly‘ℤ) ∧ ℎ ≠
0𝑝)) |
| 133 | 126, 131,
132 | 3imtr4i 292 |
. . . . . . . . . . . . 13
⊢ (ℎ ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} → ℎ ∈ ((Poly‘ℤ) ∖
{0𝑝})) |
| 134 | 133 | ssriv 3987 |
. . . . . . . . . . . 12
⊢ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖
{0𝑝}) |
| 135 | | ssrexv 4053 |
. . . . . . . . . . . . 13
⊢ ({𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖
{0𝑝}) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0 → ∃𝑐 ∈ ((Poly‘ℤ) ∖
{0𝑝})(𝑐‘𝑔) = 0)) |
| 136 | 81 | cbvrexvw 3238 |
. . . . . . . . . . . . 13
⊢
(∃𝑐 ∈
((Poly‘ℤ) ∖ {0𝑝})(𝑐‘𝑔) = 0 ↔ ∃ℎ ∈ ((Poly‘ℤ) ∖
{0𝑝})(ℎ‘𝑔) = 0) |
| 137 | 135, 136 | imbitrdi 251 |
. . . . . . . . . . . 12
⊢ ({𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} ⊆ ((Poly‘ℤ) ∖
{0𝑝}) → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0 → ∃ℎ ∈ ((Poly‘ℤ) ∖
{0𝑝})(ℎ‘𝑔) = 0)) |
| 138 | 134, 137 | ax-mp 5 |
. . . . . . . . . . 11
⊢
(∃𝑐 ∈
{𝑑 ∈
(Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧
(deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0 → ∃ℎ ∈ ((Poly‘ℤ) ∖
{0𝑝})(ℎ‘𝑔) = 0) |
| 139 | 138 | anim2i 617 |
. . . . . . . . . 10
⊢ ((𝑔 ∈ ℂ ∧
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑔) = 0) → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝})(ℎ‘𝑔) = 0)) |
| 140 | 124, 139 | sylbi 217 |
. . . . . . . . 9
⊢ (𝑔 ∈ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝})(ℎ‘𝑔) = 0)) |
| 141 | 122, 140 | biimtrdi 253 |
. . . . . . . 8
⊢ (𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝})(ℎ‘𝑔) = 0))) |
| 142 | 141 | rexlimivw 3151 |
. . . . . . 7
⊢
(∃𝑎 ∈
ℕ0 𝑓 =
{𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0} → (𝑔 ∈ 𝑓 → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝})(ℎ‘𝑔) = 0))) |
| 143 | 142 | impcom 407 |
. . . . . 6
⊢ ((𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝})(ℎ‘𝑔) = 0)) |
| 144 | 143 | exlimiv 1930 |
. . . . 5
⊢
(∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) → (𝑔 ∈ ℂ ∧ ∃ℎ ∈ ((Poly‘ℤ)
∖ {0𝑝})(ℎ‘𝑔) = 0)) |
| 145 | 121, 144 | impbii 209 |
. . . 4
⊢ ((𝑔 ∈ ℂ ∧
∃ℎ ∈
((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0) ↔ ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 146 | | elaa 26358 |
. . . 4
⊢ (𝑔 ∈ 𝔸 ↔ (𝑔 ∈ ℂ ∧
∃ℎ ∈
((Poly‘ℤ) ∖ {0𝑝})(ℎ‘𝑔) = 0)) |
| 147 | | eluniab 4921 |
. . . 4
⊢ (𝑔 ∈ ∪ {𝑓
∣ ∃𝑎 ∈
ℕ0 𝑓 =
{𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}} ↔ ∃𝑓(𝑔 ∈ 𝑓 ∧ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0})) |
| 148 | 145, 146,
147 | 3bitr4i 303 |
. . 3
⊢ (𝑔 ∈ 𝔸 ↔ 𝑔 ∈ ∪ {𝑓
∣ ∃𝑎 ∈
ℕ0 𝑓 =
{𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}}) |
| 149 | 148 | eqriv 2734 |
. 2
⊢ 𝔸
= ∪ {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}} |
| 150 | | aannenlem.a |
. . . 4
⊢ 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣
∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ)
∣ (𝑑 ≠
0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}) |
| 151 | 150 | rnmpt 5968 |
. . 3
⊢ ran 𝐻 = {𝑓 ∣ ∃𝑎 ∈ ℕ0 𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}} |
| 152 | 151 | unieqi 4919 |
. 2
⊢ ∪ ran 𝐻 = ∪ {𝑓 ∣ ∃𝑎 ∈ ℕ0
𝑓 = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝
∧ (deg‘𝑑) ≤
𝑎 ∧ ∀𝑒 ∈ ℕ0
(abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐‘𝑏) = 0}} |
| 153 | 149, 152 | eqtr4i 2768 |
1
⊢ 𝔸
= ∪ ran 𝐻 |