Step |
Hyp |
Ref |
Expression |
1 |
|
dgrle.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
2 |
|
dgrle.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
3 |
|
dgrle.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
4 |
|
dgrle.4 |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) ) |
5 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝜑 ) |
6 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝑘 ≤ 𝑁 ) |
7 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝑘 ∈ ℕ0 ) |
8 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
9 |
7 8
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝑘 ∈ ( ℤ≥ ‘ 0 ) ) |
10 |
2
|
nn0zd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
11 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝑁 ∈ ℤ ) |
12 |
|
elfz5 |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ≤ 𝑁 ) ) |
13 |
9 11 12
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ≤ 𝑁 ) ) |
14 |
6 13
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝑘 ∈ ( 0 ... 𝑁 ) ) |
15 |
5 14 3
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑘 ≤ 𝑁 ) → 𝐴 ∈ ℂ ) |
16 |
|
0cnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ ¬ 𝑘 ≤ 𝑁 ) → 0 ∈ ℂ ) |
17 |
15 16
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ∈ ℂ ) |
18 |
17
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) : ℕ0 ⟶ ℂ ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
20 |
|
eqid |
⊢ ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) = ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) |
21 |
20
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ∈ ℂ ) → ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) = if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) |
22 |
19 17 21
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) = if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) |
23 |
22
|
neeq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 ↔ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ≠ 0 ) ) |
24 |
|
iffalse |
⊢ ( ¬ 𝑘 ≤ 𝑁 → if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) = 0 ) |
25 |
24
|
necon1ai |
⊢ ( if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ≠ 0 → 𝑘 ≤ 𝑁 ) |
26 |
23 25
|
syl6bi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
27 |
26
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ0 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
28 |
|
nfv |
⊢ Ⅎ 𝑚 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) |
29 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) |
30 |
|
nfcv |
⊢ Ⅎ 𝑘 0 |
31 |
29 30
|
nfne |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 |
32 |
|
nfv |
⊢ Ⅎ 𝑘 𝑚 ≤ 𝑁 |
33 |
31 32
|
nfim |
⊢ Ⅎ 𝑘 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 → 𝑚 ≤ 𝑁 ) |
34 |
|
fveq2 |
⊢ ( 𝑘 = 𝑚 → ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) = ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ) |
35 |
34
|
neeq1d |
⊢ ( 𝑘 = 𝑚 → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 ↔ ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 ) ) |
36 |
|
breq1 |
⊢ ( 𝑘 = 𝑚 → ( 𝑘 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁 ) ) |
37 |
35 36
|
imbi12d |
⊢ ( 𝑘 = 𝑚 → ( ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ↔ ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 → 𝑚 ≤ 𝑁 ) ) ) |
38 |
28 33 37
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ ℕ0 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ↔ ∀ 𝑚 ∈ ℕ0 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 → 𝑚 ≤ 𝑁 ) ) |
39 |
27 38
|
sylib |
⊢ ( 𝜑 → ∀ 𝑚 ∈ ℕ0 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 → 𝑚 ≤ 𝑁 ) ) |
40 |
|
plyco0 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) : ℕ0 ⟶ ℂ ) → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ↔ ∀ 𝑚 ∈ ℕ0 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 → 𝑚 ≤ 𝑁 ) ) ) |
41 |
2 18 40
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ↔ ∀ 𝑚 ∈ ℕ0 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) ≠ 0 → 𝑚 ≤ 𝑁 ) ) ) |
42 |
39 41
|
mpbird |
⊢ ( 𝜑 → ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
43 |
|
nfcv |
⊢ Ⅎ 𝑚 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) |
44 |
|
nfcv |
⊢ Ⅎ 𝑘 · |
45 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑧 ↑ 𝑚 ) |
46 |
29 44 45
|
nfov |
⊢ Ⅎ 𝑘 ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) |
47 |
|
oveq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝑧 ↑ 𝑘 ) = ( 𝑧 ↑ 𝑚 ) ) |
48 |
34 47
|
oveq12d |
⊢ ( 𝑘 = 𝑚 → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) ) |
49 |
43 46 48
|
cbvsumi |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑚 ∈ ( 0 ... 𝑁 ) ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) |
50 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑁 ) → 𝑘 ∈ ℕ0 ) |
51 |
50
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
52 |
|
elfzle2 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑁 ) → 𝑘 ≤ 𝑁 ) |
53 |
52
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝑘 ≤ 𝑁 ) |
54 |
53
|
iftrued |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) = 𝐴 ) |
55 |
3
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
56 |
54 55
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ∈ ℂ ) |
57 |
51 56 21
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) = if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) |
58 |
57 54
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) = 𝐴 ) |
59 |
58
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) |
60 |
59
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) |
61 |
49 60
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑚 ∈ ( 0 ... 𝑁 ) ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) |
62 |
61
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ ℂ ↦ Σ 𝑚 ∈ ( 0 ... 𝑁 ) ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) ) |
63 |
4 62
|
eqtr4d |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑚 ∈ ( 0 ... 𝑁 ) ( ( ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) ) ) |
64 |
1 2 18 42 63
|
coeeq |
⊢ ( 𝜑 → ( coeff ‘ 𝐹 ) = ( 𝑘 ∈ ℕ0 ↦ if ( 𝑘 ≤ 𝑁 , 𝐴 , 0 ) ) ) |