| Step | Hyp | Ref
| Expression |
| 1 | | ssid 4006 |
. . . . 5
⊢ ℂ
⊆ ℂ |
| 2 | | plyconst 26245 |
. . . . 5
⊢ ((ℂ
⊆ ℂ ∧ 𝐴
∈ ℂ) → (ℂ × {𝐴}) ∈
(Poly‘ℂ)) |
| 3 | 1, 2 | mpan 690 |
. . . 4
⊢ (𝐴 ∈ ℂ → (ℂ
× {𝐴}) ∈
(Poly‘ℂ)) |
| 4 | | plyssc 26239 |
. . . . 5
⊢
(Poly‘𝑆)
⊆ (Poly‘ℂ) |
| 5 | 4 | sseli 3979 |
. . . 4
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈
(Poly‘ℂ)) |
| 6 | | plymulcl 26260 |
. . . 4
⊢
(((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ))
→ ((ℂ × {𝐴}) ∘f · 𝐹) ∈
(Poly‘ℂ)) |
| 7 | 3, 5, 6 | syl2an 596 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℂ ×
{𝐴}) ∘f
· 𝐹) ∈
(Poly‘ℂ)) |
| 8 | | eqid 2737 |
. . . 4
⊢
(coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) = (coeff‘((ℂ
× {𝐴})
∘f · 𝐹)) |
| 9 | 8 | coef3 26271 |
. . 3
⊢
(((ℂ × {𝐴}) ∘f · 𝐹) ∈ (Poly‘ℂ)
→ (coeff‘((ℂ × {𝐴}) ∘f · 𝐹)):ℕ0⟶ℂ) |
| 10 | | ffn 6736 |
. . 3
⊢
((coeff‘((ℂ × {𝐴}) ∘f · 𝐹)):ℕ0⟶ℂ →
(coeff‘((ℂ × {𝐴}) ∘f · 𝐹)) Fn
ℕ0) |
| 11 | 7, 9, 10 | 3syl 18 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ
× {𝐴})
∘f · 𝐹)) Fn ℕ0) |
| 12 | | fconstg 6795 |
. . . . 5
⊢ (𝐴 ∈ ℂ →
(ℕ0 × {𝐴}):ℕ0⟶{𝐴}) |
| 13 | 12 | adantr 480 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0
× {𝐴}):ℕ0⟶{𝐴}) |
| 14 | 13 | ffnd 6737 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℕ0
× {𝐴}) Fn
ℕ0) |
| 15 | | eqid 2737 |
. . . . . 6
⊢
(coeff‘𝐹) =
(coeff‘𝐹) |
| 16 | 15 | coef3 26271 |
. . . . 5
⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ) |
| 17 | 16 | adantl 481 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹):ℕ0⟶ℂ) |
| 18 | 17 | ffnd 6737 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘𝐹) Fn
ℕ0) |
| 19 | | nn0ex 12532 |
. . . 4
⊢
ℕ0 ∈ V |
| 20 | 19 | a1i 11 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ℕ0
∈ V) |
| 21 | | inidm 4227 |
. . 3
⊢
(ℕ0 ∩ ℕ0) =
ℕ0 |
| 22 | 14, 18, 20, 20, 21 | offn 7710 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → ((ℕ0
× {𝐴})
∘f · (coeff‘𝐹)) Fn ℕ0) |
| 23 | 3 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (ℂ
× {𝐴}) ∈
(Poly‘ℂ)) |
| 24 | | eqid 2737 |
. . . . . . 7
⊢
(coeff‘(ℂ × {𝐴})) = (coeff‘(ℂ × {𝐴})) |
| 25 | 24 | coefv0 26287 |
. . . . . 6
⊢ ((ℂ
× {𝐴}) ∈
(Poly‘ℂ) → ((ℂ × {𝐴})‘0) = ((coeff‘(ℂ ×
{𝐴}))‘0)) |
| 26 | 23, 25 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ
× {𝐴})‘0) =
((coeff‘(ℂ × {𝐴}))‘0)) |
| 27 | | simpll 767 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴 ∈
ℂ) |
| 28 | | 0cn 11253 |
. . . . . 6
⊢ 0 ∈
ℂ |
| 29 | | fvconst2g 7222 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 0 ∈
ℂ) → ((ℂ × {𝐴})‘0) = 𝐴) |
| 30 | 27, 28, 29 | sylancl 586 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → ((ℂ
× {𝐴})‘0) =
𝐴) |
| 31 | 26, 30 | eqtr3d 2779 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘(ℂ × {𝐴}))‘0) = 𝐴) |
| 32 | | simpr 484 |
. . . . . . 7
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℕ0) |
| 33 | 32 | nn0cnd 12589 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℂ) |
| 34 | 33 | subid1d 11609 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝑛 − 0) = 𝑛) |
| 35 | 34 | fveq2d 6910 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘𝐹)‘(𝑛 − 0)) = ((coeff‘𝐹)‘𝑛)) |
| 36 | 31, 35 | oveq12d 7449 |
. . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) = (𝐴 · ((coeff‘𝐹)‘𝑛))) |
| 37 | 5 | ad2antlr 727 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈
(Poly‘ℂ)) |
| 38 | 24, 15 | coemul 26291 |
. . . . 5
⊢
(((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝐹 ∈ (Poly‘ℂ)
∧ 𝑛 ∈
ℕ0) → ((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘)))) |
| 39 | 23, 37, 32, 38 | syl3anc 1373 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘)))) |
| 40 | | nn0uz 12920 |
. . . . . . 7
⊢
ℕ0 = (ℤ≥‘0) |
| 41 | 32, 40 | eleqtrdi 2851 |
. . . . . 6
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
(ℤ≥‘0)) |
| 42 | | fzss2 13604 |
. . . . . 6
⊢ (𝑛 ∈
(ℤ≥‘0) → (0...0) ⊆ (0...𝑛)) |
| 43 | 41, 42 | syl 17 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (0...0)
⊆ (0...𝑛)) |
| 44 | | elfz1eq 13575 |
. . . . . . . 8
⊢ (𝑘 ∈ (0...0) → 𝑘 = 0) |
| 45 | 44 | adantl 481 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) → 𝑘 = 0) |
| 46 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 0 →
((coeff‘(ℂ × {𝐴}))‘𝑘) = ((coeff‘(ℂ × {𝐴}))‘0)) |
| 47 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑘 = 0 → (𝑛 − 𝑘) = (𝑛 − 0)) |
| 48 | 47 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑘 = 0 → ((coeff‘𝐹)‘(𝑛 − 𝑘)) = ((coeff‘𝐹)‘(𝑛 − 0))) |
| 49 | 46, 48 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑘 = 0 →
(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) ·
((coeff‘𝐹)‘(𝑛 − 0)))) |
| 50 | 45, 49 | syl 17 |
. . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) ·
((coeff‘𝐹)‘(𝑛 − 0)))) |
| 51 | 17 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘𝐹)‘𝑛) ∈ ℂ) |
| 52 | 27, 51 | mulcld 11281 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → (𝐴 · ((coeff‘𝐹)‘𝑛)) ∈ ℂ) |
| 53 | 36, 52 | eqeltrd 2841 |
. . . . . . 7
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈
ℂ) |
| 54 | 53 | adantr 480 |
. . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) →
(((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈
ℂ) |
| 55 | 50, 54 | eqeltrd 2841 |
. . . . 5
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...0)) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) ∈ ℂ) |
| 56 | | eldifn 4132 |
. . . . . . . . 9
⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → ¬ 𝑘 ∈
(0...0)) |
| 57 | 56 | adantl 481 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → ¬ 𝑘 ∈
(0...0)) |
| 58 | | eldifi 4131 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ (0...𝑛)) |
| 59 | | elfznn0 13660 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0) |
| 60 | 58, 59 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → 𝑘 ∈ ℕ0) |
| 61 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(deg‘(ℂ × {𝐴})) = (deg‘(ℂ × {𝐴})) |
| 62 | 24, 61 | dgrub 26273 |
. . . . . . . . . . . . 13
⊢
(((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0
∧ ((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(ℂ × {𝐴}))) |
| 63 | 62 | 3expia 1122 |
. . . . . . . . . . . 12
⊢
(((ℂ × {𝐴}) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0)
→ (((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴})))) |
| 64 | 23, 60, 63 | syl2an 596 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(ℂ × {𝐴})))) |
| 65 | | 0dgr 26284 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℂ →
(deg‘(ℂ × {𝐴})) = 0) |
| 66 | 65 | ad3antrrr 730 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
(deg‘(ℂ × {𝐴})) = 0) |
| 67 | 66 | breq2d 5155 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ
× {𝐴})) ↔ 𝑘 ≤ 0)) |
| 68 | 60 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → 𝑘 ∈ ℕ0) |
| 69 | | nn0le0eq0 12554 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ (𝑘 ≤ 0 ↔
𝑘 = 0)) |
| 70 | 68, 69 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ 0 ↔ 𝑘 = 0)) |
| 71 | 67, 70 | bitrd 279 |
. . . . . . . . . . 11
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (𝑘 ≤ (deg‘(ℂ
× {𝐴})) ↔ 𝑘 = 0)) |
| 72 | 64, 71 | sylibd 239 |
. . . . . . . . . 10
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 = 0)) |
| 73 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑘 = 0 → 𝑘 = 0) |
| 74 | | 0z 12624 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℤ |
| 75 | | elfz3 13574 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℤ → 0 ∈ (0...0)) |
| 76 | 74, 75 | ax-mp 5 |
. . . . . . . . . . 11
⊢ 0 ∈
(0...0) |
| 77 | 73, 76 | eqeltrdi 2849 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → 𝑘 ∈ (0...0)) |
| 78 | 72, 77 | syl6 35 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) ≠ 0 → 𝑘 ∈ (0...0))) |
| 79 | 78 | necon1bd 2958 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (¬ 𝑘 ∈ (0...0) →
((coeff‘(ℂ × {𝐴}))‘𝑘) = 0)) |
| 80 | 57, 79 | mpd 15 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
((coeff‘(ℂ × {𝐴}))‘𝑘) = 0) |
| 81 | 80 | oveq1d 7446 |
. . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (0 · ((coeff‘𝐹)‘(𝑛 − 𝑘)))) |
| 82 | 17 | adantr 480 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(coeff‘𝐹):ℕ0⟶ℂ) |
| 83 | | fznn0sub 13596 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈
ℕ0) |
| 84 | 58, 83 | syl 17 |
. . . . . . . 8
⊢ (𝑘 ∈ ((0...𝑛) ∖ (0...0)) → (𝑛 − 𝑘) ∈
ℕ0) |
| 85 | | ffvelcdm 7101 |
. . . . . . . 8
⊢
(((coeff‘𝐹):ℕ0⟶ℂ ∧
(𝑛 − 𝑘) ∈ ℕ0)
→ ((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ) |
| 86 | 82, 84, 85 | syl2an 596 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
((coeff‘𝐹)‘(𝑛 − 𝑘)) ∈ ℂ) |
| 87 | 86 | mul02d 11459 |
. . . . . 6
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) → (0 ·
((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0) |
| 88 | 81, 87 | eqtrd 2777 |
. . . . 5
⊢ ((((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ ((0...𝑛) ∖ (0...0))) →
(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = 0) |
| 89 | | fzfid 14014 |
. . . . 5
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(0...𝑛) ∈
Fin) |
| 90 | 43, 55, 88, 89 | fsumss 15761 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
Σ𝑘 ∈
(0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑛)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘)))) |
| 91 | 49 | fsum1 15783 |
. . . . 5
⊢ ((0
∈ ℤ ∧ (((coeff‘(ℂ × {𝐴}))‘0) · ((coeff‘𝐹)‘(𝑛 − 0))) ∈ ℂ) →
Σ𝑘 ∈
(0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) ·
((coeff‘𝐹)‘(𝑛 − 0)))) |
| 92 | 74, 53, 91 | sylancr 587 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
Σ𝑘 ∈
(0...0)(((coeff‘(ℂ × {𝐴}))‘𝑘) · ((coeff‘𝐹)‘(𝑛 − 𝑘))) = (((coeff‘(ℂ × {𝐴}))‘0) ·
((coeff‘𝐹)‘(𝑛 − 0)))) |
| 93 | 39, 90, 92 | 3eqtr2d 2783 |
. . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = (((coeff‘(ℂ × {𝐴}))‘0) ·
((coeff‘𝐹)‘(𝑛 − 0)))) |
| 94 | | simpl 482 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐴 ∈ ℂ) |
| 95 | | eqidd 2738 |
. . . 4
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘𝐹)‘𝑛) = ((coeff‘𝐹)‘𝑛)) |
| 96 | 20, 94, 18, 95 | ofc1 7725 |
. . 3
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(((ℕ0 × {𝐴}) ∘f ·
(coeff‘𝐹))‘𝑛) = (𝐴 · ((coeff‘𝐹)‘𝑛))) |
| 97 | 36, 93, 96 | 3eqtr4d 2787 |
. 2
⊢ (((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
((coeff‘((ℂ × {𝐴}) ∘f · 𝐹))‘𝑛) = (((ℕ0 × {𝐴}) ∘f ·
(coeff‘𝐹))‘𝑛)) |
| 98 | 11, 22, 97 | eqfnfvd 7054 |
1
⊢ ((𝐴 ∈ ℂ ∧ 𝐹 ∈ (Poly‘𝑆)) → (coeff‘((ℂ
× {𝐴})
∘f · 𝐹)) = ((ℕ0 × {𝐴}) ∘f ·
(coeff‘𝐹))) |