Step | Hyp | Ref
| Expression |
1 | | plymulcl 23781 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 · 𝐺) ∈
(Poly‘ℂ)) |
2 | | coeadd.3 |
. . . . 5
⊢ 𝑀 = (deg‘𝐹) |
3 | | dgrcl 23793 |
. . . . 5
⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈
ℕ0) |
4 | 2, 3 | syl5eqel 2692 |
. . . 4
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑀 ∈
ℕ0) |
5 | | coeadd.4 |
. . . . 5
⊢ 𝑁 = (deg‘𝐺) |
6 | | dgrcl 23793 |
. . . . 5
⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈
ℕ0) |
7 | 5, 6 | syl5eqel 2692 |
. . . 4
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝑁 ∈
ℕ0) |
8 | | nn0addcl 11205 |
. . . 4
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝑀 + 𝑁) ∈
ℕ0) |
9 | 4, 7, 8 | syl2an 493 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑀 + 𝑁) ∈
ℕ0) |
10 | | fzfid 12634 |
. . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
(0...𝑛) ∈
Fin) |
11 | | coefv0.1 |
. . . . . . . . . 10
⊢ 𝐴 = (coeff‘𝐹) |
12 | 11 | coef3 23792 |
. . . . . . . . 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 12302 |
. . . . . . 7
⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ0) |
16 | | ffvelrn 6265 |
. . . . . . 7
⊢ ((𝐴:ℕ0⟶ℂ ∧
𝑘 ∈
ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
17 | 14, 15, 16 | syl2an 493 |
. . . . . 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 23792 |
. . . . . . . . 9
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐵:ℕ0⟶ℂ) |
20 | 19 | adantl 481 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐵:ℕ0⟶ℂ) |
21 | 20 | ad2antrr 758 |
. . . . . . 7
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → 𝐵:ℕ0⟶ℂ) |
22 | | fznn0sub 12244 |
. . . . . . . 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 | ffvelrnd 6268 |
. . . . . 6
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → (𝐵‘(𝑛 − 𝑘)) ∈ ℂ) |
25 | 17, 24 | mulcld 9939 |
. . . . 5
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ) |
26 | 10, 25 | fsumcl 14311 |
. . . 4
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑛 ∈ ℕ0) →
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) ∈ ℂ) |
27 | | eqid 2610 |
. . . 4
⊢ (𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) = (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) |
28 | 26, 27 | fmptd 6292 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))):ℕ0⟶ℂ) |
29 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗)) |
30 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑗 → (𝑛 − 𝑘) = (𝑗 − 𝑘)) |
31 | 30 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑗 → (𝐵‘(𝑛 − 𝑘)) = (𝐵‘(𝑗 − 𝑘))) |
32 | 31 | oveq2d 6565 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑗 → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
33 | 32 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ (0...𝑛)) → ((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
34 | 29, 33 | sumeq12dv 14284 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
35 | | sumex 14266 |
. . . . . . . . . 10
⊢
Σ𝑘 ∈
(0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) ∈ V |
36 | 34, 27, 35 | fvmpt 6191 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘)))) |
37 | 36 | ad2antrl 760 |
. . . . . . . 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)))) |
38 | | simp2r 1081 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ 𝑗 ≤ (𝑀 + 𝑁)) |
39 | | simp2l 1080 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℕ0) |
40 | 39 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑗 ∈ ℝ) |
41 | | simp3l 1082 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ (0...𝑗)) |
42 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ0) |
43 | 41, 42 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℕ0) |
44 | 43 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ∈ ℝ) |
45 | 7 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑁 ∈
ℕ0) |
46 | 45 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈
ℕ0) |
47 | 46 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑁 ∈ ℝ) |
48 | 40, 44, 47 | lesubadd2d 10505 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 − 𝑘) ≤ 𝑁 ↔ 𝑗 ≤ (𝑘 + 𝑁))) |
49 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝑀 ∈
ℕ0) |
50 | 49 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈
ℕ0) |
51 | 50 | nn0red 11229 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑀 ∈ ℝ) |
52 | | simp3r 1083 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝑘 ≤ 𝑀) |
53 | 44, 51, 47, 52 | leadd1dd 10520 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) |
54 | 44, 47 | readdcld 9948 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑘 + 𝑁) ∈ ℝ) |
55 | 51, 47 | readdcld 9948 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑀 + 𝑁) ∈ ℝ) |
56 | | letr 10010 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑗 ∈ ℝ ∧ (𝑘 + 𝑁) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → ((𝑗 ≤ (𝑘 + 𝑁) ∧ (𝑘 + 𝑁) ≤ (𝑀 + 𝑁)) → 𝑗 ≤ (𝑀 + 𝑁))) |
57 | 40, 54, 55, 56 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . 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))) |
58 | 53, 57 | mpan2d 706 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 ≤ (𝑘 + 𝑁) → 𝑗 ≤ (𝑀 + 𝑁))) |
59 | 48, 58 | sylbid 229 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝑗 − 𝑘) ≤ 𝑁 → 𝑗 ≤ (𝑀 + 𝑁))) |
60 | 38, 59 | mtod 188 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ¬ (𝑗 − 𝑘) ≤ 𝑁) |
61 | | simpr 476 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 ∈ (Poly‘𝑆)) |
62 | 61 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐺 ∈ (Poly‘𝑆)) |
63 | | fznn0sub 12244 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (0...𝑗) → (𝑗 − 𝑘) ∈
ℕ0) |
64 | 41, 63 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝑗 − 𝑘) ∈
ℕ0) |
65 | 18, 5 | dgrub 23794 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ0 ∧ (𝐵‘(𝑗 − 𝑘)) ≠ 0) → (𝑗 − 𝑘) ≤ 𝑁) |
66 | 65 | 3expia 1259 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐺 ∈ (Poly‘𝑆) ∧ (𝑗 − 𝑘) ∈ ℕ0) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁)) |
67 | 62, 64, 66 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐵‘(𝑗 − 𝑘)) ≠ 0 → (𝑗 − 𝑘) ≤ 𝑁)) |
68 | 67 | necon1bd 2800 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (¬ (𝑗 − 𝑘) ≤ 𝑁 → (𝐵‘(𝑗 − 𝑘)) = 0)) |
69 | 60, 68 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐵‘(𝑗 − 𝑘)) = 0) |
70 | 69 | oveq2d 6565 |
. . . . . . . . . . . . . 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)) |
71 | 13 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → 𝐴:ℕ0⟶ℂ) |
72 | 71, 43 | ffvelrnd 6268 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → (𝐴‘𝑘) ∈ ℂ) |
73 | 72 | mul01d 10114 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · 0) = 0) |
74 | 70, 73 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁)) ∧ (𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
75 | 74 | 3expia 1259 |
. . . . . . . . . . . 12
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → ((𝑘 ∈ (0...𝑗) ∧ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0)) |
76 | 75 | impl 648 |
. . . . . . . . . . 11
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
77 | | simpl 472 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 ∈ (Poly‘𝑆)) |
78 | 77 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → 𝐹 ∈ (Poly‘𝑆)) |
79 | 11, 2 | dgrub 23794 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0 ∧ (𝐴‘𝑘) ≠ 0) → 𝑘 ≤ 𝑀) |
80 | 79 | 3expia 1259 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)) |
81 | 78, 42, 80 | syl2an 493 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑀)) |
82 | 81 | necon1bd 2800 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → (¬ 𝑘 ≤ 𝑀 → (𝐴‘𝑘) = 0)) |
83 | 82 | imp 444 |
. . . . . . . . . . . . 13
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐴‘𝑘) = 0) |
84 | 83 | oveq1d 6564 |
. . . . . . . . . . . 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)))) |
85 | 20 | ad3antrrr 762 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → 𝐵:ℕ0⟶ℂ) |
86 | 63 | ad2antlr 759 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝑗 − 𝑘) ∈
ℕ0) |
87 | 85, 86 | ffvelrnd 6268 |
. . . . . . . . . . . . 13
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (𝐵‘(𝑗 − 𝑘)) ∈ ℂ) |
88 | 87 | mul02d 10113 |
. . . . . . . . . . . 12
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → (0 · (𝐵‘(𝑗 − 𝑘))) = 0) |
89 | 84, 88 | eqtrd 2644 |
. . . . . . . . . . 11
⊢
(((((𝐹 ∈
(Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) ∧ ¬ 𝑘 ≤ 𝑀) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
90 | 76, 89 | pm2.61dan 828 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
91 | 90 | sumeq2dv 14281 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = Σ𝑘 ∈ (0...𝑗)0) |
92 | | fzfi 12633 |
. . . . . . . . . . 11
⊢
(0...𝑗) ∈
Fin |
93 | 92 | olci 405 |
. . . . . . . . . 10
⊢
((0...𝑗) ⊆
(ℤ≥‘0) ∨ (0...𝑗) ∈ Fin) |
94 | | sumz 14300 |
. . . . . . . . . 10
⊢
(((0...𝑗) ⊆
(ℤ≥‘0) ∨ (0...𝑗) ∈ Fin) → Σ𝑘 ∈ (0...𝑗)0 = 0) |
95 | 93, 94 | ax-mp 5 |
. . . . . . . . 9
⊢
Σ𝑘 ∈
(0...𝑗)0 =
0 |
96 | 91, 95 | syl6eq 2660 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ (𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ (𝑀 + 𝑁))) → Σ𝑘 ∈ (0...𝑗)((𝐴‘𝑘) · (𝐵‘(𝑗 − 𝑘))) = 0) |
97 | 37, 96 | eqtrd 2644 |
. . . . . . 7
⊢ (((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 | expr 641 |
. . . . . 6
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (¬
𝑗 ≤ (𝑀 + 𝑁) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) = 0)) |
99 | 98 | necon1ad 2799 |
. . . . 5
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ ℕ0) → (((𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))) |
100 | 99 | ralrimiva 2949 |
. . . 4
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ∀𝑗 ∈ ℕ0 (((𝑛 ∈ ℕ0
↦ Σ𝑘 ∈
(0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ≠ 0 → 𝑗 ≤ (𝑀 + 𝑁))) |
101 | | plyco0 23752 |
. . . . 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)))) |
102 | 9, 28, 101 | syl2anc 691 |
. . . 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)))) |
103 | 100, 102 | mpbird 246 |
. . 3
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘)))) “
(ℤ≥‘((𝑀 + 𝑁) + 1))) = {0}) |
104 | 11, 2 | dgrub2 23795 |
. . . . . 6
⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “
(ℤ≥‘(𝑀 + 1))) = {0}) |
105 | 104 | adantr 480 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐴 “
(ℤ≥‘(𝑀 + 1))) = {0}) |
106 | 18, 5 | dgrub2 23795 |
. . . . . 6
⊢ (𝐺 ∈ (Poly‘𝑆) → (𝐵 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
107 | 106 | adantl 481 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐵 “
(ℤ≥‘(𝑁 + 1))) = {0}) |
108 | 11, 2 | coeid 23798 |
. . . . . 6
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
109 | 108 | adantr 480 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑀)((𝐴‘𝑘) · (𝑧↑𝑘)))) |
110 | 18, 5 | coeid 23798 |
. . . . . 6
⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) |
111 | 110 | adantl 481 |
. . . . 5
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐵‘𝑘) · (𝑧↑𝑘)))) |
112 | 77, 61, 49, 45, 13, 20, 105, 107, 109, 111 | plymullem1 23774 |
. . . 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)))) |
113 | | elfznn0 12302 |
. . . . . . . 8
⊢ (𝑗 ∈ (0...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ0) |
114 | 113, 36 | 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)))) |
115 | 114 | oveq1d 6564 |
. . . . . 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))) |
116 | 115 | sumeq2i 14277 |
. . . . 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)) |
117 | 116 | mpteq2i 4669 |
. . . 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))) |
118 | 112, 117 | syl6eqr 2662 |
. . 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)))) |
119 | 1, 9, 28, 103, 118 | coeeq 23787 |
. 2
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (coeff‘(𝐹 ∘𝑓 · 𝐺)) = (𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))) |
120 | | ffvelrn 6265 |
. . . 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) |
121 | 28, 113, 120 | syl2an 493 |
. . 3
⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) ∧ 𝑗 ∈ (0...(𝑀 + 𝑁))) → ((𝑛 ∈ ℕ0 ↦
Σ𝑘 ∈ (0...𝑛)((𝐴‘𝑘) · (𝐵‘(𝑛 − 𝑘))))‘𝑗) ∈ ℂ) |
122 | 1, 9, 121, 118 | dgrle 23803 |
. 2
⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (deg‘(𝐹 ∘𝑓 · 𝐺)) ≤ (𝑀 + 𝑁)) |
123 | 119, 122 | jca 553 |
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))) |