| Step | Hyp | Ref
| Expression |
| 1 | | plymulcl 26260 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) ∈
(Poly‘ℂ)) |
| 2 | | coeadd.3 |
. . . . 5
⊢ 𝑀 = (deg‘𝐹) |
| 3 | | dgrcl 26272 |
. . . . 5
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈
ℕ0) |
| 4 | 2, 3 | eqeltrid 2845 |
. . . 4
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈
ℕ0) |
| 5 | | coeadd.4 |
. . . . 5
⊢ 𝑁 = (deg‘𝐺) |
| 6 | | dgrcl 26272 |
. . . . 5
⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈
ℕ0) |
| 7 | 5, 6 | eqeltrid 2845 |
. . . 4
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈
ℕ0) |
| 8 | | nn0addcl 12561 |
. . . 4
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝑀 + 𝑁) ∈
ℕ0) |
| 9 | 4, 7, 8 | syl2an 596 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈
ℕ0) |
| 10 | | fzfid 14014 |
. . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(0...𝑛) ∈
Fin) |
| 11 | | coefv0.1 |
. . . . . . . . . 10
⊢ 𝐴 = (coeff‘𝐹) |
| 12 | 11 | coef3 26271 |
. . . . . . . . 9
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
| 13 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐴:ℕ0⟶ℂ) |
| 14 | 13 | adantr 480 |
. . . . . . 7
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) → 𝐴:ℕ0⟶ℂ) |
| 15 | | elfznn0 13660 |
. . . . . . 7
⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0) |
| 16 | | ffvelcdm 7101 |
. . . . . . 7
⊢ ((𝐴:ℕ0⟶ℂ ∧
𝑘 ∈
ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
| 17 | 14, 15, 16 | syl2an 596 |
. . . . . 6
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴‘𝑘) ∈ ℂ) |
| 18 | | coeadd.2 |
. . . . . . . . . 10
⊢ 𝐵 = (coeff‘𝐺) |
| 19 | 18 | coef3 26271 |
. . . . . . . . 9
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ) |
| 20 | 19 | adantl 481 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ) |
| 21 | 20 | ad2antrr 726 |
. . . . . . 7
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ0⟶ℂ) |
| 22 | | fznn0sub 13596 |
. . . . . . . 8
⊢ (𝑘 ∈ (0...𝑛) → (𝑛 − 𝑘) ∈
ℕ0) |
| 23 | 22 | adantl 481 |
. . . . . . 7
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝑛 − 𝑘) ∈
ℕ0) |
| 24 | 21, 23 | ffvelcdmd 7105 |
. . . . . 6
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ) |
| 25 | 17, 24 | mulcld 11281 |
. . . . 5
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ) |
| 26 | 10, 25 | fsumcl 15769 |
. . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ) |
| 27 | 26 | fmpttd 7135 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ0⟶ℂ) |
| 28 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗)) |
| 29 | | fvoveq1 7454 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑗 → (𝐵‘(𝑛 − 𝑘)) = (𝐵‘(𝑗 − 𝑘))) |
| 30 | 29 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑗 → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
| 31 | 30 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
| 32 | 28, 31 | sumeq12dv 15742 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
| 33 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) = (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) |
| 34 | | sumex 15724 |
. . . . . . . . . 10
⊢
Σ𝑘 ∈
(0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) ∈ V |
| 35 | 32, 33, 34 | fvmpt 7016 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
| 36 | 35 | ad2antrl 728 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
| 37 | | simp2r 1201 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ 𝑗 ≤ (𝑀 + 𝑁)) |
| 38 | | simp2l 1200 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℕ0) |
| 39 | 38 | nn0red 12588 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℝ) |
| 40 | | simp3l 1202 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ (0...𝑗)) |
| 41 | | elfznn0 13660 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ0) |
| 42 | 40, 41 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℕ0) |
| 43 | 42 | nn0red 12588 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℝ) |
| 44 | 7 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈
ℕ0) |
| 45 | 44 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈
ℕ0) |
| 46 | 45 | nn0red 12588 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℝ) |
| 47 | 39, 43, 46 | lesubadd2d 11862 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 − 𝑘) ≤ 𝑁 ↔ 𝑗 ≤ (𝑘 + 𝑁))) |
| 48 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈
ℕ0) |
| 49 | 48 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈
ℕ0) |
| 50 | 49 | nn0red 12588 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈ ℝ) |
| 51 | | simp3r 1203 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ≤ 𝑀) |
| 52 | 43, 50, 46, 51 | leadd1dd 11877 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) |
| 53 | 43, 46 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ∈ ℝ) |
| 54 | 50, 46 | readdcld 11290 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑀 + 𝑁) ∈ ℝ) |
| 55 | | letr 11355 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑗 ∈ ℝ ∧ (𝑘 + 𝑁) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁))) |
| 56 | 39, 53, 54, 55 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁))) |
| 57 | 52, 56 | mpan2d 694 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 ≤ (𝑘 + 𝑁) → 𝑗 ≤ (𝑀 + 𝑁))) |
| 58 | 47, 57 | sylbid 240 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 − 𝑘) ≤ 𝑁 → 𝑗 ≤ (𝑀 + 𝑁))) |
| 59 | 37, 58 | mtod 198 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ (𝑗 − 𝑘) ≤ 𝑁) |
| 60 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆)) |
| 61 | 60 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐺 ∈ (Poly‘𝑆)) |
| 62 | | fznn0sub 13596 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (0...𝑗) → (𝑗 − 𝑘) ∈
ℕ0) |
| 63 | 40, 62 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 − 𝑘) ∈
ℕ0) |
| 64 | 18, 5 | dgrub 26273 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ0 ∧ (𝐵‘(𝑗 − 𝑘)) ≠ 0) → (𝑗 − 𝑘) ≤ 𝑁) |
| 65 | 64 | 3expia 1122 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ0) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁)) |
| 66 | 61, 63, 65 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁)) |
| 67 | 66 | necon1bd 2958 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (¬ (𝑗 − 𝑘) ≤ 𝑁 → (𝐵‘(𝑗 − 𝑘)) = 0)) |
| 68 | 59, 67 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐵‘(𝑗 − 𝑘)) = 0) |
| 69 | 68 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = ((𝐴‘𝑘) · 0)) |
| 70 | 13 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐴:ℕ0⟶ℂ) |
| 71 | 70, 42 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐴‘𝑘) ∈ ℂ) |
| 72 | 71 | mul01d 11460 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · 0) = 0) |
| 73 | 69, 72 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
| 74 | 73 | 3expia 1122 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)) |
| 75 | 74 | impl 455 |
. . . . . . . . . . 11
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
| 76 | | simpl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) |
| 77 | 76 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆)) |
| 78 | 11, 2 | dgrub 26273 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀) |
| 79 | 78 | 3expia 1122 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)) |
| 80 | 77, 41, 79 | syl2an 596 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)) |
| 81 | 80 | necon1bd 2958 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0)) |
| 82 | 81 | imp 406 |
. . . . . . . . . . . . 13
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0) |
| 83 | 82 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = (0 · (𝐵‘(𝑗 − 𝑘)))) |
| 84 | 20 | ad3antrrr 730 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ0⟶ℂ) |
| 85 | 62 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝑗 − 𝑘) ∈
ℕ0) |
| 86 | 84, 85 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘(𝑗 − 𝑘)) ∈ ℂ) |
| 87 | 86 | mul02d 11459 |
. . . . . . . . . . . 12
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘(𝑗 − 𝑘))) = 0) |
| 88 | 83, 87 | eqtrd 2777 |
. . . . . . . . . . 11
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
| 89 | 75, 88 | pm2.61dan 813 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
| 90 | 89 | sumeq2dv 15738 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)0) |
| 91 | | fzfi 14013 |
. . . . . . . . . . 11
⊢
(0...𝑗) ∈
Fin |
| 92 | 91 | olci 867 |
. . . . . . . . . 10
⊢
((0...𝑗) ⊆
(ℤ≥‘0) ∨ (0...𝑗) ∈ Fin) |
| 93 | | sumz 15758 |
. . . . . . . . . 10
⊢
(((0...𝑗) ⊆
(ℤ≥‘0) ∨ (0...𝑗) ∈ Fin) → Σ𝑘 ∈ (0...𝑗)0 = 0) |
| 94 | 92, 93 | ax-mp 5 |
. . . . . . . . 9
⊢
Σ𝑘 ∈
(0...𝑗)0 =
0 |
| 95 | 90, 94 | eqtrdi 2793 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
| 96 | 36, 95 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = 0) |
| 97 | 96 | expr 456 |
. . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (¬
𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = 0)) |
| 98 | 97 | necon1ad 2957 |
. . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))) |
| 99 | 98 | ralrimiva 3146 |
. . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))) |
| 100 | | plyco0 26231 |
. . . . 5
⊢ (((𝑀 + 𝑁) ∈ ℕ0 ∧ (𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ0⟶ℂ)
→ (((𝑛 ∈
ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “
(ℤ≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑗 ∈ ℕ0
(((𝑛 ∈
ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))) |
| 101 | 9, 27, 100 | syl2anc 584 |
. . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “
(ℤ≥‘((𝑀 + 𝑁) + 1))) = {0} ↔ ∀𝑗 ∈ ℕ0
(((𝑛 ∈
ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁)))) |
| 102 | 99, 101 | mpbird 257 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “
(ℤ≥‘((𝑀 + 𝑁) + 1))) = {0}) |
| 103 | 11, 2 | dgrub2 26274 |
. . . . . 6
⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “
(ℤ≥‘(𝑀 + 1))) = {0}) |
| 104 | 103 | adantr 480 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “
(ℤ≥‘(𝑀 + 1))) = {0}) |
| 105 | 18, 5 | dgrub2 26274 |
. . . . . 6
⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
| 106 | 105 | adantl 481 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
| 107 | 11, 2 | coeid 26277 |
. . . . . 6
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
| 108 | 107 | adantr 480 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
| 109 | 18, 5 | coeid 26277 |
. . . . . 6
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) |
| 110 | 109 | adantl 481 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) |
| 111 | 76, 60, 48, 44, 13, 20, 104, 106, 108, 110 | plymullem1 26253 |
. . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗)))) |
| 112 | | elfznn0 13660 |
. . . . . . . 8
⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ0) |
| 113 | 112, 35 | syl 17 |
. . . . . . 7
⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
| 114 | 113 | oveq1d 7446 |
. . . . . 6
⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → (((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗)) = (Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗))) |
| 115 | 114 | sumeq2i 15734 |
. . . . 5
⊢
Σ𝑗 ∈
(0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗)) = Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗)) |
| 116 | 115 | mpteq2i 5247 |
. . . 4
⊢ (𝑧 ∈ ℂ ↦
Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗))) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) · (𝑧↑𝑗))) |
| 117 | 111, 116 | eqtr4di 2795 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑗 ∈ (0...(𝑀 + 𝑁))(((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) · (𝑧↑𝑗)))) |
| 118 | 1, 9, 27, 102, 117 | coeeq 26266 |
. 2
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘f · 𝐺)) = (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))) |
| 119 | | ffvelcdm 7101 |
. . . 4
⊢ (((𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ0⟶ℂ ∧
𝑗 ∈
ℕ0) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ∈ ℂ) |
| 120 | 27, 112, 119 | syl2an 596 |
. . 3
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ∈ ℂ) |
| 121 | 1, 9, 120, 117 | dgrle 26282 |
. 2
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘f · 𝐺)) ≤ (𝑀 + 𝑁)) |
| 122 | 118, 121 | jca 511 |
1
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((coeff‘(𝐹 ∘f · 𝐺)) = (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) ∧ (deg‘(𝐹 ∘f · 𝐺)) ≤ (𝑀 + 𝑁))) |