Step |
Hyp |
Ref |
Expression |
1 |
|
coefv0.1 |
⊢ 𝐴 = ( coeff ‘ 𝐹 ) |
2 |
|
coeadd.2 |
⊢ 𝐵 = ( coeff ‘ 𝐺 ) |
3 |
|
coeadd.3 |
⊢ 𝑀 = ( deg ‘ 𝐹 ) |
4 |
|
coeadd.4 |
⊢ 𝑁 = ( deg ‘ 𝐺 ) |
5 |
|
plymulcl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f · 𝐺 ) ∈ ( Poly ‘ ℂ ) ) |
6 |
|
dgrcl |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( deg ‘ 𝐹 ) ∈ ℕ0 ) |
7 |
3 6
|
eqeltrid |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑀 ∈ ℕ0 ) |
8 |
|
dgrcl |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → ( deg ‘ 𝐺 ) ∈ ℕ0 ) |
9 |
4 8
|
eqeltrid |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → 𝑁 ∈ ℕ0 ) |
10 |
|
nn0addcl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 + 𝑁 ) ∈ ℕ0 ) |
11 |
7 9 10
|
syl2an |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝑀 + 𝑁 ) ∈ ℕ0 ) |
12 |
|
fzfid |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 0 ... 𝑛 ) ∈ Fin ) |
13 |
1
|
coef3 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐴 : ℕ0 ⟶ ℂ ) |
14 |
13
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐴 : ℕ0 ⟶ ℂ ) |
15 |
14
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) → 𝐴 : ℕ0 ⟶ ℂ ) |
16 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑛 ) → 𝑘 ∈ ℕ0 ) |
17 |
|
ffvelrn |
⊢ ( ( 𝐴 : ℕ0 ⟶ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
18 |
15 16 17
|
syl2an |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
19 |
2
|
coef3 |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → 𝐵 : ℕ0 ⟶ ℂ ) |
20 |
19
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐵 : ℕ0 ⟶ ℂ ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → 𝐵 : ℕ0 ⟶ ℂ ) |
22 |
|
fznn0sub |
⊢ ( 𝑘 ∈ ( 0 ... 𝑛 ) → ( 𝑛 − 𝑘 ) ∈ ℕ0 ) |
23 |
22
|
adantl |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑛 − 𝑘 ) ∈ ℕ0 ) |
24 |
21 23
|
ffvelrnd |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ∈ ℂ ) |
25 |
18 24
|
mulcld |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ∈ ℂ ) |
26 |
12 25
|
fsumcl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑛 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ∈ ℂ ) |
27 |
26
|
fmpttd |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) : ℕ0 ⟶ ℂ ) |
28 |
|
oveq2 |
⊢ ( 𝑛 = 𝑗 → ( 0 ... 𝑛 ) = ( 0 ... 𝑗 ) ) |
29 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑗 → ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) = ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝑛 = 𝑗 → ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) = ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝑛 = 𝑗 ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) = ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ) |
32 |
28 31
|
sumeq12dv |
⊢ ( 𝑛 = 𝑗 → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ) |
33 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) = ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) |
34 |
|
sumex |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ∈ V |
35 |
32 33 34
|
fvmpt |
⊢ ( 𝑗 ∈ ℕ0 → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ) |
36 |
35
|
ad2antrl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ) |
37 |
|
simp2r |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) |
38 |
|
simp2l |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑗 ∈ ℕ0 ) |
39 |
38
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑗 ∈ ℝ ) |
40 |
|
simp3l |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑘 ∈ ( 0 ... 𝑗 ) ) |
41 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑗 ) → 𝑘 ∈ ℕ0 ) |
42 |
40 41
|
syl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
43 |
42
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑘 ∈ ℝ ) |
44 |
9
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝑁 ∈ ℕ0 ) |
45 |
44
|
3ad2ant1 |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑁 ∈ ℕ0 ) |
46 |
45
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑁 ∈ ℝ ) |
47 |
39 43 46
|
lesubadd2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( ( 𝑗 − 𝑘 ) ≤ 𝑁 ↔ 𝑗 ≤ ( 𝑘 + 𝑁 ) ) ) |
48 |
7
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝑀 ∈ ℕ0 ) |
49 |
48
|
3ad2ant1 |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑀 ∈ ℕ0 ) |
50 |
49
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑀 ∈ ℝ ) |
51 |
|
simp3r |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝑘 ≤ 𝑀 ) |
52 |
43 50 46 51
|
leadd1dd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( 𝑘 + 𝑁 ) ≤ ( 𝑀 + 𝑁 ) ) |
53 |
43 46
|
readdcld |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( 𝑘 + 𝑁 ) ∈ ℝ ) |
54 |
50 46
|
readdcld |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( 𝑀 + 𝑁 ) ∈ ℝ ) |
55 |
|
letr |
⊢ ( ( 𝑗 ∈ ℝ ∧ ( 𝑘 + 𝑁 ) ∈ ℝ ∧ ( 𝑀 + 𝑁 ) ∈ ℝ ) → ( ( 𝑗 ≤ ( 𝑘 + 𝑁 ) ∧ ( 𝑘 + 𝑁 ) ≤ ( 𝑀 + 𝑁 ) ) → 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) |
56 |
39 53 54 55
|
syl3anc |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ¬ ( 𝑗 − 𝑘 ) ≤ 𝑁 ) |
60 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐺 ∈ ( Poly ‘ 𝑆 ) ) |
61 |
60
|
3ad2ant1 |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝐺 ∈ ( Poly ‘ 𝑆 ) ) |
62 |
|
fznn0sub |
⊢ ( 𝑘 ∈ ( 0 ... 𝑗 ) → ( 𝑗 − 𝑘 ) ∈ ℕ0 ) |
63 |
40 62
|
syl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( 𝑗 − 𝑘 ) ∈ ℕ0 ) |
64 |
2 4
|
dgrub |
⊢ ( ( 𝐺 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑗 − 𝑘 ) ∈ ℕ0 ∧ ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ≠ 0 ) → ( 𝑗 − 𝑘 ) ≤ 𝑁 ) |
65 |
64
|
3expia |
⊢ ( ( 𝐺 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑗 − 𝑘 ) ∈ ℕ0 ) → ( ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ≠ 0 → ( 𝑗 − 𝑘 ) ≤ 𝑁 ) ) |
66 |
61 63 65
|
syl2anc |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) = 0 ) |
69 |
68
|
oveq2d |
⊢ ( ( ( 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 |
14
|
3ad2ant1 |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → 𝐴 : ℕ0 ⟶ ℂ ) |
71 |
70 42
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
72 |
71
|
mul01d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ∧ ( 𝑘 ∈ ( 0 ... 𝑗 ) ∧ 𝑘 ≤ 𝑀 ) ) → ( ( 𝐴 ‘ 𝑘 ) · 0 ) = 0 ) |
73 |
69 72
|
eqtrd |
⊢ ( ( ( 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 |
⊢ ( ( ( 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 |
⊢ ( ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) ∧ 𝑘 ≤ 𝑀 ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) = 0 ) |
76 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
77 |
76
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
78 |
1 3
|
dgrub |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑘 ∈ ℕ0 ∧ ( 𝐴 ‘ 𝑘 ) ≠ 0 ) → 𝑘 ≤ 𝑀 ) |
79 |
78
|
3expia |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
80 |
77 41 79
|
syl2an |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
81 |
80
|
necon1bd |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) → ( ¬ 𝑘 ≤ 𝑀 → ( 𝐴 ‘ 𝑘 ) = 0 ) ) |
82 |
81
|
imp |
⊢ ( ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) ∧ ¬ 𝑘 ≤ 𝑀 ) → ( 𝐴 ‘ 𝑘 ) = 0 ) |
83 |
82
|
oveq1d |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) ∧ ¬ 𝑘 ≤ 𝑀 ) → 𝐵 : ℕ0 ⟶ ℂ ) |
85 |
62
|
ad2antlr |
⊢ ( ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) ∧ ¬ 𝑘 ≤ 𝑀 ) → ( 𝑗 − 𝑘 ) ∈ ℕ0 ) |
86 |
84 85
|
ffvelrnd |
⊢ ( ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) ∧ ¬ 𝑘 ≤ 𝑀 ) → ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ∈ ℂ ) |
87 |
86
|
mul02d |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑗 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) = 0 ) |
90 |
89
|
sumeq2dv |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) → Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝑗 ) 0 ) |
91 |
|
fzfi |
⊢ ( 0 ... 𝑗 ) ∈ Fin |
92 |
91
|
olci |
⊢ ( ( 0 ... 𝑗 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 𝑗 ) ∈ Fin ) |
93 |
|
sumz |
⊢ ( ( ( 0 ... 𝑗 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 𝑗 ) ∈ Fin ) → Σ 𝑘 ∈ ( 0 ... 𝑗 ) 0 = 0 ) |
94 |
92 93
|
ax-mp |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑗 ) 0 = 0 |
95 |
90 94
|
eqtrdi |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) → Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) = 0 ) |
96 |
36 95
|
eqtrd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑗 ∈ ℕ0 ∧ ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) = 0 ) |
97 |
96
|
expr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑗 ∈ ℕ0 ) → ( ¬ 𝑗 ≤ ( 𝑀 + 𝑁 ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) = 0 ) ) |
98 |
97
|
necon1ad |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑗 ∈ ℕ0 ) → ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) ≠ 0 → 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) |
99 |
98
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ∀ 𝑗 ∈ ℕ0 ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) ≠ 0 → 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) |
100 |
|
plyco0 |
⊢ ( ( ( 𝑀 + 𝑁 ) ∈ ℕ0 ∧ ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) : ℕ0 ⟶ ℂ ) → ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) “ ( ℤ≥ ‘ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = { 0 } ↔ ∀ 𝑗 ∈ ℕ0 ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) ≠ 0 → 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ) |
101 |
11 27 100
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) “ ( ℤ≥ ‘ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = { 0 } ↔ ∀ 𝑗 ∈ ℕ0 ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) ≠ 0 → 𝑗 ≤ ( 𝑀 + 𝑁 ) ) ) ) |
102 |
99 101
|
mpbird |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) “ ( ℤ≥ ‘ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = { 0 } ) |
103 |
1 3
|
dgrub2 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
104 |
103
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
105 |
2 4
|
dgrub2 |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
106 |
105
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
107 |
1 3
|
coeid |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
108 |
107
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
109 |
2 4
|
coeid |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
110 |
109
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
111 |
76 60 48 44 14 20 104 106 108 110
|
plymullem1 |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f · 𝐺 ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑗 ) ) ) ) |
112 |
|
elfznn0 |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → 𝑗 ∈ ℕ0 ) |
113 |
112 35
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) = Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) ) |
114 |
113
|
oveq1d |
⊢ ( 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) · ( 𝑧 ↑ 𝑗 ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑗 ) ) ) |
115 |
114
|
sumeq2i |
⊢ Σ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) · ( 𝑧 ↑ 𝑗 ) ) = Σ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑗 ) ) |
116 |
115
|
mpteq2i |
⊢ ( 𝑧 ∈ ℂ ↦ Σ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) · ( 𝑧 ↑ 𝑗 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑗 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑗 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑗 ) ) ) |
117 |
111 116
|
eqtr4di |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f · 𝐺 ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) · ( 𝑧 ↑ 𝑗 ) ) ) ) |
118 |
5 11 27 102 117
|
coeeq |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( coeff ‘ ( 𝐹 ∘f · 𝐺 ) ) = ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ) |
119 |
|
ffvelrn |
⊢ ( ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) : ℕ0 ⟶ ℂ ∧ 𝑗 ∈ ℕ0 ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) ∈ ℂ ) |
120 |
27 112 119
|
syl2an |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑗 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ‘ 𝑗 ) ∈ ℂ ) |
121 |
5 11 120 117
|
dgrle |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( deg ‘ ( 𝐹 ∘f · 𝐺 ) ) ≤ ( 𝑀 + 𝑁 ) ) |
122 |
118 121
|
jca |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( coeff ‘ ( 𝐹 ∘f · 𝐺 ) ) = ( 𝑛 ∈ ℕ0 ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ) ∧ ( deg ‘ ( 𝐹 ∘f · 𝐺 ) ) ≤ ( 𝑀 + 𝑁 ) ) ) |