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 |
|
plyaddcl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f + 𝐺 ) ∈ ( Poly ‘ ℂ ) ) |
6 |
|
dgrcl |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → ( deg ‘ 𝐺 ) ∈ ℕ0 ) |
7 |
4 6
|
eqeltrid |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → 𝑁 ∈ ℕ0 ) |
8 |
7
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝑁 ∈ ℕ0 ) |
9 |
|
dgrcl |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( deg ‘ 𝐹 ) ∈ ℕ0 ) |
10 |
3 9
|
eqeltrid |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑀 ∈ ℕ0 ) |
11 |
10
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝑀 ∈ ℕ0 ) |
12 |
8 11
|
ifcld |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℕ0 ) |
13 |
|
addcl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + 𝑦 ) ∈ ℂ ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 + 𝑦 ) ∈ ℂ ) |
15 |
1
|
coef3 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐴 : ℕ0 ⟶ ℂ ) |
16 |
15
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐴 : ℕ0 ⟶ ℂ ) |
17 |
2
|
coef3 |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → 𝐵 : ℕ0 ⟶ ℂ ) |
18 |
17
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐵 : ℕ0 ⟶ ℂ ) |
19 |
|
nn0ex |
⊢ ℕ0 ∈ V |
20 |
19
|
a1i |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ℕ0 ∈ V ) |
21 |
|
inidm |
⊢ ( ℕ0 ∩ ℕ0 ) = ℕ0 |
22 |
14 16 18 20 20 21
|
off |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐴 ∘f + 𝐵 ) : ℕ0 ⟶ ℂ ) |
23 |
|
oveq12 |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) = 0 ∧ ( 𝐵 ‘ 𝑘 ) = 0 ) → ( ( 𝐴 ‘ 𝑘 ) + ( 𝐵 ‘ 𝑘 ) ) = ( 0 + 0 ) ) |
24 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
25 |
23 24
|
eqtrdi |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) = 0 ∧ ( 𝐵 ‘ 𝑘 ) = 0 ) → ( ( 𝐴 ‘ 𝑘 ) + ( 𝐵 ‘ 𝑘 ) ) = 0 ) |
26 |
16
|
ffnd |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐴 Fn ℕ0 ) |
27 |
18
|
ffnd |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐵 Fn ℕ0 ) |
28 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) ) |
29 |
|
eqidd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) ) |
30 |
26 27 20 20 21 28 29
|
ofval |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) + ( 𝐵 ‘ 𝑘 ) ) ) |
31 |
30
|
eqeq1d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) = 0 ↔ ( ( 𝐴 ‘ 𝑘 ) + ( 𝐵 ‘ 𝑘 ) ) = 0 ) ) |
32 |
25 31
|
syl5ibr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ‘ 𝑘 ) = 0 ∧ ( 𝐵 ‘ 𝑘 ) = 0 ) → ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) = 0 ) ) |
33 |
32
|
necon3ad |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ≠ 0 → ¬ ( ( 𝐴 ‘ 𝑘 ) = 0 ∧ ( 𝐵 ‘ 𝑘 ) = 0 ) ) ) |
34 |
|
neorian |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) ≠ 0 ∨ ( 𝐵 ‘ 𝑘 ) ≠ 0 ) ↔ ¬ ( ( 𝐴 ‘ 𝑘 ) = 0 ∧ ( 𝐵 ‘ 𝑘 ) = 0 ) ) |
35 |
33 34
|
syl6ibr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ≠ 0 → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 ∨ ( 𝐵 ‘ 𝑘 ) ≠ 0 ) ) ) |
36 |
1 3
|
dgrub2 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
37 |
36
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
38 |
|
plyco0 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐴 : ℕ0 ⟶ ℂ ) → ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) ) |
39 |
11 16 38
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) ) |
40 |
37 39
|
mpbid |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ∀ 𝑘 ∈ ℕ0 ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
41 |
40
|
r19.21bi |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
42 |
11
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑀 ∈ ℕ0 ) |
43 |
42
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑀 ∈ ℝ ) |
44 |
8
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
45 |
44
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑁 ∈ ℝ ) |
46 |
|
max1 |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) |
47 |
43 45 46
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) |
48 |
|
nn0re |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℝ ) |
50 |
12
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℕ0 ) |
51 |
50
|
nn0red |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℝ ) |
52 |
|
letr |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℝ ) → ( ( 𝑘 ≤ 𝑀 ∧ 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
53 |
49 43 51 52
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 ≤ 𝑀 ∧ 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
54 |
47 53
|
mpan2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ≤ 𝑀 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
55 |
41 54
|
syld |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
56 |
2 4
|
dgrub2 |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
57 |
56
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
58 |
|
plyco0 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐵 : ℕ0 ⟶ ℂ ) → ( ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) ) |
59 |
8 18 58
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) ) |
60 |
57 59
|
mpbid |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ∀ 𝑘 ∈ ℕ0 ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
61 |
60
|
r19.21bi |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
62 |
|
max2 |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) |
63 |
43 45 62
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) |
64 |
|
letr |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℝ ) → ( ( 𝑘 ≤ 𝑁 ∧ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
65 |
49 45 51 64
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 ≤ 𝑁 ∧ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
66 |
63 65
|
mpan2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ≤ 𝑁 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
67 |
61 66
|
syld |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
68 |
55 67
|
jaod |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ‘ 𝑘 ) ≠ 0 ∨ ( 𝐵 ‘ 𝑘 ) ≠ 0 ) → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
69 |
35 68
|
syld |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
70 |
69
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ∀ 𝑘 ∈ ℕ0 ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |
71 |
|
plyco0 |
⊢ ( ( if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℕ0 ∧ ( 𝐴 ∘f + 𝐵 ) : ℕ0 ⟶ ℂ ) → ( ( ( 𝐴 ∘f + 𝐵 ) “ ( ℤ≥ ‘ ( if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) ) |
72 |
12 22 71
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( ( 𝐴 ∘f + 𝐵 ) “ ( ℤ≥ ‘ ( if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) ) |
73 |
70 72
|
mpbird |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( 𝐴 ∘f + 𝐵 ) “ ( ℤ≥ ‘ ( if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) + 1 ) ) ) = { 0 } ) |
74 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
75 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐺 ∈ ( Poly ‘ 𝑆 ) ) |
76 |
1 3
|
coeid |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
77 |
76
|
adantr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
78 |
2 4
|
coeid |
⊢ ( 𝐺 ∈ ( Poly ‘ 𝑆 ) → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
79 |
78
|
adantl |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
80 |
74 75 11 8 16 18 37 57 77 79
|
plyaddlem1 |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( 𝐹 ∘f + 𝐺 ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ( ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
81 |
5 12 22 73 80
|
coeeq |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( coeff ‘ ( 𝐹 ∘f + 𝐺 ) ) = ( 𝐴 ∘f + 𝐵 ) ) |
82 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
83 |
|
ffvelrn |
⊢ ( ( ( 𝐴 ∘f + 𝐵 ) : ℕ0 ⟶ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ∈ ℂ ) |
84 |
22 82 83
|
syl2an |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) ∧ 𝑘 ∈ ( 0 ... if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) → ( ( 𝐴 ∘f + 𝐵 ) ‘ 𝑘 ) ∈ ℂ ) |
85 |
5 12 84 80
|
dgrle |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( deg ‘ ( 𝐹 ∘f + 𝐺 ) ) ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) |
86 |
81 85
|
jca |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( coeff ‘ ( 𝐹 ∘f + 𝐺 ) ) = ( 𝐴 ∘f + 𝐵 ) ∧ ( deg ‘ ( 𝐹 ∘f + 𝐺 ) ) ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) |