Step |
Hyp |
Ref |
Expression |
1 |
|
eldifi |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → 𝐹 ∈ ( Poly ‘ ℝ ) ) |
2 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
3 |
|
1re |
⊢ 1 ∈ ℝ |
4 |
|
plyid |
⊢ ( ( ℝ ⊆ ℂ ∧ 1 ∈ ℝ ) → Xp ∈ ( Poly ‘ ℝ ) ) |
5 |
2 3 4
|
mp2an |
⊢ Xp ∈ ( Poly ‘ ℝ ) |
6 |
5
|
a1i |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → Xp ∈ ( Poly ‘ ℝ ) ) |
7 |
|
simprl |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → 𝑥 ∈ ℝ ) |
8 |
|
simprr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → 𝑦 ∈ ℝ ) |
9 |
7 8
|
readdcld |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 + 𝑦 ) ∈ ℝ ) |
10 |
7 8
|
remulcld |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ) → ( 𝑥 · 𝑦 ) ∈ ℝ ) |
11 |
1 6 9 10
|
plymul |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( 𝐹 ∘f · Xp ) ∈ ( Poly ‘ ℝ ) ) |
12 |
|
0re |
⊢ 0 ∈ ℝ |
13 |
|
eqid |
⊢ ( coeff ‘ ( 𝐹 ∘f · Xp ) ) = ( coeff ‘ ( 𝐹 ∘f · Xp ) ) |
14 |
13
|
coef2 |
⊢ ( ( ( 𝐹 ∘f · Xp ) ∈ ( Poly ‘ ℝ ) ∧ 0 ∈ ℝ ) → ( coeff ‘ ( 𝐹 ∘f · Xp ) ) : ℕ0 ⟶ ℝ ) |
15 |
11 12 14
|
sylancl |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( coeff ‘ ( 𝐹 ∘f · Xp ) ) : ℕ0 ⟶ ℝ ) |
16 |
15
|
feqmptd |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( coeff ‘ ( 𝐹 ∘f · Xp ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( coeff ‘ ( 𝐹 ∘f · Xp ) ) ‘ 𝑛 ) ) ) |
17 |
|
cnex |
⊢ ℂ ∈ V |
18 |
17
|
a1i |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ℂ ∈ V ) |
19 |
|
plyf |
⊢ ( 𝐹 ∈ ( Poly ‘ ℝ ) → 𝐹 : ℂ ⟶ ℂ ) |
20 |
1 19
|
syl |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → 𝐹 : ℂ ⟶ ℂ ) |
21 |
|
plyf |
⊢ ( Xp ∈ ( Poly ‘ ℝ ) → Xp : ℂ ⟶ ℂ ) |
22 |
5 21
|
ax-mp |
⊢ Xp : ℂ ⟶ ℂ |
23 |
22
|
a1i |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → Xp : ℂ ⟶ ℂ ) |
24 |
|
simprl |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → 𝑥 ∈ ℂ ) |
25 |
|
simprr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → 𝑦 ∈ ℂ ) |
26 |
24 25
|
mulcomd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 · 𝑦 ) = ( 𝑦 · 𝑥 ) ) |
27 |
18 20 23 26
|
caofcom |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( 𝐹 ∘f · Xp ) = ( Xp ∘f · 𝐹 ) ) |
28 |
27
|
fveq2d |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( coeff ‘ ( 𝐹 ∘f · Xp ) ) = ( coeff ‘ ( Xp ∘f · 𝐹 ) ) ) |
29 |
28
|
fveq1d |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( ( coeff ‘ ( 𝐹 ∘f · Xp ) ) ‘ 𝑛 ) = ( ( coeff ‘ ( Xp ∘f · 𝐹 ) ) ‘ 𝑛 ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → ( ( coeff ‘ ( 𝐹 ∘f · Xp ) ) ‘ 𝑛 ) = ( ( coeff ‘ ( Xp ∘f · 𝐹 ) ) ‘ 𝑛 ) ) |
31 |
5
|
a1i |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → Xp ∈ ( Poly ‘ ℝ ) ) |
32 |
1
|
adantr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → 𝐹 ∈ ( Poly ‘ ℝ ) ) |
33 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℕ0 ) |
34 |
|
eqid |
⊢ ( coeff ‘ Xp ) = ( coeff ‘ Xp ) |
35 |
|
eqid |
⊢ ( coeff ‘ 𝐹 ) = ( coeff ‘ 𝐹 ) |
36 |
34 35
|
coemul |
⊢ ( ( Xp ∈ ( Poly ‘ ℝ ) ∧ 𝐹 ∈ ( Poly ‘ ℝ ) ∧ 𝑛 ∈ ℕ0 ) → ( ( coeff ‘ ( Xp ∘f · 𝐹 ) ) ‘ 𝑛 ) = Σ 𝑖 ∈ ( 0 ... 𝑛 ) ( ( ( coeff ‘ Xp ) ‘ 𝑖 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) |
37 |
31 32 33 36
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → ( ( coeff ‘ ( Xp ∘f · 𝐹 ) ) ‘ 𝑛 ) = Σ 𝑖 ∈ ( 0 ... 𝑛 ) ( ( ( coeff ‘ Xp ) ‘ 𝑖 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) |
38 |
|
elfznn0 |
⊢ ( 𝑖 ∈ ( 0 ... 𝑛 ) → 𝑖 ∈ ℕ0 ) |
39 |
|
coeidp |
⊢ ( 𝑖 ∈ ℕ0 → ( ( coeff ‘ Xp ) ‘ 𝑖 ) = if ( 𝑖 = 1 , 1 , 0 ) ) |
40 |
38 39
|
syl |
⊢ ( 𝑖 ∈ ( 0 ... 𝑛 ) → ( ( coeff ‘ Xp ) ‘ 𝑖 ) = if ( 𝑖 = 1 , 1 , 0 ) ) |
41 |
40
|
oveq1d |
⊢ ( 𝑖 ∈ ( 0 ... 𝑛 ) → ( ( ( coeff ‘ Xp ) ‘ 𝑖 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = ( if ( 𝑖 = 1 , 1 , 0 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) |
42 |
|
ovif |
⊢ ( if ( 𝑖 = 1 , 1 , 0 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) |
43 |
41 42
|
eqtrdi |
⊢ ( 𝑖 ∈ ( 0 ... 𝑛 ) → ( ( ( coeff ‘ Xp ) ‘ 𝑖 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) ) |
44 |
43
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( ( ( coeff ‘ Xp ) ‘ 𝑖 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) ) |
45 |
44
|
sumeq2dv |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) ( ( ( coeff ‘ Xp ) ‘ 𝑖 ) · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) ) |
46 |
|
velsn |
⊢ ( 𝑖 ∈ { 1 } ↔ 𝑖 = 1 ) |
47 |
46
|
bicomi |
⊢ ( 𝑖 = 1 ↔ 𝑖 ∈ { 1 } ) |
48 |
47
|
a1i |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( 𝑖 = 1 ↔ 𝑖 ∈ { 1 } ) ) |
49 |
35
|
coef2 |
⊢ ( ( 𝐹 ∈ ( Poly ‘ ℝ ) ∧ 0 ∈ ℝ ) → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ℝ ) |
50 |
1 12 49
|
sylancl |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ℝ ) |
51 |
50
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ℝ ) |
52 |
|
fznn0sub |
⊢ ( 𝑖 ∈ ( 0 ... 𝑛 ) → ( 𝑛 − 𝑖 ) ∈ ℕ0 ) |
53 |
52
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( 𝑛 − 𝑖 ) ∈ ℕ0 ) |
54 |
51 53
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ∈ ℝ ) |
55 |
54
|
recnd |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ∈ ℂ ) |
56 |
55
|
mulid2d |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) |
57 |
55
|
mul02d |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) = 0 ) |
58 |
48 56 57
|
ifbieq12d |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑖 ∈ ( 0 ... 𝑛 ) ) → if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) = if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) ) |
59 |
58
|
sumeq2dv |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) = Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) ) |
60 |
|
eqeq2 |
⊢ ( 0 = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) → ( Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = 0 ↔ Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) ) |
61 |
|
eqeq2 |
⊢ ( ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) → ( Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ↔ Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) ) |
62 |
|
oveq2 |
⊢ ( 𝑛 = 0 → ( 0 ... 𝑛 ) = ( 0 ... 0 ) ) |
63 |
|
0z |
⊢ 0 ∈ ℤ |
64 |
|
fzsn |
⊢ ( 0 ∈ ℤ → ( 0 ... 0 ) = { 0 } ) |
65 |
63 64
|
ax-mp |
⊢ ( 0 ... 0 ) = { 0 } |
66 |
62 65
|
eqtrdi |
⊢ ( 𝑛 = 0 → ( 0 ... 𝑛 ) = { 0 } ) |
67 |
|
elsni |
⊢ ( 𝑖 ∈ { 0 } → 𝑖 = 0 ) |
68 |
67
|
adantl |
⊢ ( ( 𝑛 = 0 ∧ 𝑖 ∈ { 0 } ) → 𝑖 = 0 ) |
69 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
70 |
69
|
nesymi |
⊢ ¬ 0 = 1 |
71 |
|
eqeq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 = 1 ↔ 0 = 1 ) ) |
72 |
70 71
|
mtbiri |
⊢ ( 𝑖 = 0 → ¬ 𝑖 = 1 ) |
73 |
68 72
|
syl |
⊢ ( ( 𝑛 = 0 ∧ 𝑖 ∈ { 0 } ) → ¬ 𝑖 = 1 ) |
74 |
47
|
notbii |
⊢ ( ¬ 𝑖 = 1 ↔ ¬ 𝑖 ∈ { 1 } ) |
75 |
74
|
biimpi |
⊢ ( ¬ 𝑖 = 1 → ¬ 𝑖 ∈ { 1 } ) |
76 |
|
iffalse |
⊢ ( ¬ 𝑖 ∈ { 1 } → if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = 0 ) |
77 |
73 75 76
|
3syl |
⊢ ( ( 𝑛 = 0 ∧ 𝑖 ∈ { 0 } ) → if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = 0 ) |
78 |
66 77
|
sumeq12rdv |
⊢ ( 𝑛 = 0 → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = Σ 𝑖 ∈ { 0 } 0 ) |
79 |
|
snfi |
⊢ { 0 } ∈ Fin |
80 |
79
|
olci |
⊢ ( { 0 } ⊆ ( ℤ≥ ‘ 0 ) ∨ { 0 } ∈ Fin ) |
81 |
|
sumz |
⊢ ( ( { 0 } ⊆ ( ℤ≥ ‘ 0 ) ∨ { 0 } ∈ Fin ) → Σ 𝑖 ∈ { 0 } 0 = 0 ) |
82 |
80 81
|
ax-mp |
⊢ Σ 𝑖 ∈ { 0 } 0 = 0 |
83 |
78 82
|
eqtrdi |
⊢ ( 𝑛 = 0 → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = 0 ) |
84 |
83
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ 𝑛 = 0 ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = 0 ) |
85 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ) |
86 |
33
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ0 ) |
87 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → ¬ 𝑛 = 0 ) |
88 |
87
|
neqned |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ≠ 0 ) |
89 |
|
elnnne0 |
⊢ ( 𝑛 ∈ ℕ ↔ ( 𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0 ) ) |
90 |
86 88 89
|
sylanbrc |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → 𝑛 ∈ ℕ ) |
91 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
92 |
91
|
a1i |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → 1 ∈ ℕ0 ) |
93 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ ) |
94 |
93
|
nnnn0d |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → 𝑛 ∈ ℕ0 ) |
95 |
93
|
nnge1d |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → 1 ≤ 𝑛 ) |
96 |
|
elfz2nn0 |
⊢ ( 1 ∈ ( 0 ... 𝑛 ) ↔ ( 1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛 ) ) |
97 |
92 94 95 96
|
syl3anbrc |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → 1 ∈ ( 0 ... 𝑛 ) ) |
98 |
97
|
snssd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → { 1 } ⊆ ( 0 ... 𝑛 ) ) |
99 |
50
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑖 ∈ { 1 } ) → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ℝ ) |
100 |
|
oveq2 |
⊢ ( 𝑖 = 1 → ( 𝑛 − 𝑖 ) = ( 𝑛 − 1 ) ) |
101 |
46 100
|
sylbi |
⊢ ( 𝑖 ∈ { 1 } → ( 𝑛 − 𝑖 ) = ( 𝑛 − 1 ) ) |
102 |
101
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑖 ∈ { 1 } ) → ( 𝑛 − 𝑖 ) = ( 𝑛 − 1 ) ) |
103 |
|
nnm1nn0 |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 − 1 ) ∈ ℕ0 ) |
104 |
103
|
ad2antlr |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑖 ∈ { 1 } ) → ( 𝑛 − 1 ) ∈ ℕ0 ) |
105 |
102 104
|
eqeltrd |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑖 ∈ { 1 } ) → ( 𝑛 − 𝑖 ) ∈ ℕ0 ) |
106 |
99 105
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑖 ∈ { 1 } ) → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ∈ ℝ ) |
107 |
106
|
recnd |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) ∧ 𝑖 ∈ { 1 } ) → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ∈ ℂ ) |
108 |
107
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → ∀ 𝑖 ∈ { 1 } ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ∈ ℂ ) |
109 |
|
fzfi |
⊢ ( 0 ... 𝑛 ) ∈ Fin |
110 |
109
|
olci |
⊢ ( ( 0 ... 𝑛 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 𝑛 ) ∈ Fin ) |
111 |
110
|
a1i |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → ( ( 0 ... 𝑛 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 𝑛 ) ∈ Fin ) ) |
112 |
|
sumss2 |
⊢ ( ( ( { 1 } ⊆ ( 0 ... 𝑛 ) ∧ ∀ 𝑖 ∈ { 1 } ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ∈ ℂ ) ∧ ( ( 0 ... 𝑛 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 𝑛 ) ∈ Fin ) ) → Σ 𝑖 ∈ { 1 } ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) = Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) ) |
113 |
98 108 111 112
|
syl21anc |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → Σ 𝑖 ∈ { 1 } ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) = Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) ) |
114 |
50
|
adantr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ℝ ) |
115 |
103
|
adantl |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → ( 𝑛 − 1 ) ∈ ℕ0 ) |
116 |
114 115
|
ffvelrnd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ∈ ℝ ) |
117 |
116
|
recnd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ∈ ℂ ) |
118 |
100
|
fveq2d |
⊢ ( 𝑖 = 1 → ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) |
119 |
118
|
sumsn |
⊢ ( ( 1 ∈ ℝ ∧ ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ∈ ℂ ) → Σ 𝑖 ∈ { 1 } ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) |
120 |
3 117 119
|
sylancr |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → Σ 𝑖 ∈ { 1 } ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) |
121 |
113 120
|
eqtr3d |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) |
122 |
85 90 121
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) ∧ ¬ 𝑛 = 0 ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) |
123 |
60 61 84 122
|
ifbothda |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 ∈ { 1 } , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) , 0 ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) |
124 |
59 123
|
eqtrd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → Σ 𝑖 ∈ ( 0 ... 𝑛 ) if ( 𝑖 = 1 , ( 1 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) , ( 0 · ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 𝑖 ) ) ) ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) |
125 |
37 45 124
|
3eqtrd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → ( ( coeff ‘ ( Xp ∘f · 𝐹 ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) |
126 |
30 125
|
eqtrd |
⊢ ( ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) ∧ 𝑛 ∈ ℕ0 ) → ( ( coeff ‘ ( 𝐹 ∘f · Xp ) ) ‘ 𝑛 ) = if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) |
127 |
126
|
mpteq2dva |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( 𝑛 ∈ ℕ0 ↦ ( ( coeff ‘ ( 𝐹 ∘f · Xp ) ) ‘ 𝑛 ) ) = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) ) |
128 |
16 127
|
eqtrd |
⊢ ( 𝐹 ∈ ( ( Poly ‘ ℝ ) ∖ { 0𝑝 } ) → ( coeff ‘ ( 𝐹 ∘f · Xp ) ) = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 0 , 0 , ( ( coeff ‘ 𝐹 ) ‘ ( 𝑛 − 1 ) ) ) ) ) |