Step |
Hyp |
Ref |
Expression |
1 |
|
coeeq.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
2 |
|
coeeq.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
3 |
|
coeeq.3 |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
4 |
|
coeeq.4 |
⊢ ( 𝜑 → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
5 |
|
coeeq.5 |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
6 |
|
coeval |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( coeff ‘ 𝐹 ) = ( ℩ 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ) |
7 |
1 6
|
syl |
⊢ ( 𝜑 → ( coeff ‘ 𝐹 ) = ( ℩ 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ) |
8 |
|
fvoveq1 |
⊢ ( 𝑛 = 𝑁 → ( ℤ≥ ‘ ( 𝑛 + 1 ) ) = ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) |
9 |
8
|
imaeq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = ( 𝐴 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ↔ ( 𝐴 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) ) |
11 |
|
oveq2 |
⊢ ( 𝑛 = 𝑁 → ( 0 ... 𝑛 ) = ( 0 ... 𝑁 ) ) |
12 |
11
|
sumeq1d |
⊢ ( 𝑛 = 𝑁 → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
13 |
12
|
mpteq2dv |
⊢ ( 𝑛 = 𝑁 → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
14 |
13
|
eqeq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ↔ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
15 |
10 14
|
anbi12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ↔ ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ) |
16 |
15
|
rspcev |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ∃ 𝑛 ∈ ℕ0 ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
17 |
2 4 5 16
|
syl12anc |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ0 ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
18 |
|
cnex |
⊢ ℂ ∈ V |
19 |
|
nn0ex |
⊢ ℕ0 ∈ V |
20 |
18 19
|
elmap |
⊢ ( 𝐴 ∈ ( ℂ ↑m ℕ0 ) ↔ 𝐴 : ℕ0 ⟶ ℂ ) |
21 |
3 20
|
sylibr |
⊢ ( 𝜑 → 𝐴 ∈ ( ℂ ↑m ℕ0 ) ) |
22 |
|
coeeu |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ∃! 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
23 |
1 22
|
syl |
⊢ ( 𝜑 → ∃! 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
24 |
|
imaeq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) ) |
25 |
24
|
eqeq1d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ↔ ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ) ) |
26 |
|
fveq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) ) |
27 |
26
|
oveq1d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
28 |
27
|
sumeq2sdv |
⊢ ( 𝑎 = 𝐴 → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
29 |
28
|
mpteq2dv |
⊢ ( 𝑎 = 𝐴 → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
30 |
29
|
eqeq2d |
⊢ ( 𝑎 = 𝐴 → ( 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ↔ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
31 |
25 30
|
anbi12d |
⊢ ( 𝑎 = 𝐴 → ( ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ↔ ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ) |
32 |
31
|
rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ↔ ∃ 𝑛 ∈ ℕ0 ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ) |
33 |
32
|
riota2 |
⊢ ( ( 𝐴 ∈ ( ℂ ↑m ℕ0 ) ∧ ∃! 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( ∃ 𝑛 ∈ ℕ0 ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ↔ ( ℩ 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) = 𝐴 ) ) |
34 |
21 23 33
|
syl2anc |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ℕ0 ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ↔ ( ℩ 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) = 𝐴 ) ) |
35 |
17 34
|
mpbid |
⊢ ( 𝜑 → ( ℩ 𝑎 ∈ ( ℂ ↑m ℕ0 ) ∃ 𝑛 ∈ ℕ0 ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) = 𝐴 ) |
36 |
7 35
|
eqtrd |
⊢ ( 𝜑 → ( coeff ‘ 𝐹 ) = 𝐴 ) |