Step |
Hyp |
Ref |
Expression |
1 |
|
dgrval.1 |
⊢ 𝐴 = ( coeff ‘ 𝐹 ) |
2 |
|
elply2 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ↔ ( 𝑆 ⊆ ℂ ∧ ∃ 𝑛 ∈ ℕ0 ∃ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ) |
3 |
2
|
simprbi |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ∃ 𝑛 ∈ ℕ0 ∃ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
4 |
|
simplrr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) |
5 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
6 |
|
plybss |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑆 ⊆ ℂ ) |
7 |
5 6
|
syl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝑆 ⊆ ℂ ) |
8 |
|
0cnd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 0 ∈ ℂ ) |
9 |
8
|
snssd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → { 0 } ⊆ ℂ ) |
10 |
7 9
|
unssd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( 𝑆 ∪ { 0 } ) ⊆ ℂ ) |
11 |
|
cnex |
⊢ ℂ ∈ V |
12 |
|
ssexg |
⊢ ( ( ( 𝑆 ∪ { 0 } ) ⊆ ℂ ∧ ℂ ∈ V ) → ( 𝑆 ∪ { 0 } ) ∈ V ) |
13 |
10 11 12
|
sylancl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( 𝑆 ∪ { 0 } ) ∈ V ) |
14 |
|
nn0ex |
⊢ ℕ0 ∈ V |
15 |
|
elmapg |
⊢ ( ( ( 𝑆 ∪ { 0 } ) ∈ V ∧ ℕ0 ∈ V ) → ( 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ↔ 𝑎 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
16 |
13 14 15
|
sylancl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ↔ 𝑎 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
17 |
4 16
|
mpbid |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝑎 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
18 |
|
simplrl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝑛 ∈ ℕ0 ) |
19 |
17 10
|
fssd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝑎 : ℕ0 ⟶ ℂ ) |
20 |
|
simprl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ) |
21 |
|
simprr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
22 |
5 18 19 20 21
|
coeeq |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( coeff ‘ 𝐹 ) = 𝑎 ) |
23 |
1 22
|
eqtr2id |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝑎 = 𝐴 ) |
24 |
23
|
feq1d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( 𝑎 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ↔ 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
25 |
17 24
|
mpbid |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
26 |
25
|
ex |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) → ( ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) → 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
27 |
26
|
rexlimdvva |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( ∃ 𝑛 ∈ ℕ0 ∃ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) → 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) ) |
28 |
3 27
|
mpd |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
29 |
|
nn0ssz |
⊢ ℕ0 ⊆ ℤ |
30 |
|
ffn |
⊢ ( 𝑎 : ℕ0 ⟶ ℂ → 𝑎 Fn ℕ0 ) |
31 |
|
elpreima |
⊢ ( 𝑎 Fn ℕ0 → ( 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ↔ ( 𝑥 ∈ ℕ0 ∧ ( 𝑎 ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) ) |
32 |
19 30 31
|
3syl |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ↔ ( 𝑥 ∈ ℕ0 ∧ ( 𝑎 ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) ) |
33 |
32
|
biimpa |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ∧ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ) → ( 𝑥 ∈ ℕ0 ∧ ( 𝑎 ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) |
34 |
|
eldifsni |
⊢ ( ( 𝑎 ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) → ( 𝑎 ‘ 𝑥 ) ≠ 0 ) |
35 |
33 34
|
simpl2im |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ∧ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ) → ( 𝑎 ‘ 𝑥 ) ≠ 0 ) |
36 |
33
|
simpld |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ∧ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ) → 𝑥 ∈ ℕ0 ) |
37 |
|
plyco0 |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝑎 : ℕ0 ⟶ ℂ ) → ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ↔ ∀ 𝑥 ∈ ℕ0 ( ( 𝑎 ‘ 𝑥 ) ≠ 0 → 𝑥 ≤ 𝑛 ) ) ) |
38 |
18 19 37
|
syl2anc |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ↔ ∀ 𝑥 ∈ ℕ0 ( ( 𝑎 ‘ 𝑥 ) ≠ 0 → 𝑥 ≤ 𝑛 ) ) ) |
39 |
20 38
|
mpbid |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ∀ 𝑥 ∈ ℕ0 ( ( 𝑎 ‘ 𝑥 ) ≠ 0 → 𝑥 ≤ 𝑛 ) ) |
40 |
39
|
r19.21bi |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ∧ 𝑥 ∈ ℕ0 ) → ( ( 𝑎 ‘ 𝑥 ) ≠ 0 → 𝑥 ≤ 𝑛 ) ) |
41 |
36 40
|
syldan |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ∧ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ) → ( ( 𝑎 ‘ 𝑥 ) ≠ 0 → 𝑥 ≤ 𝑛 ) ) |
42 |
35 41
|
mpd |
⊢ ( ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) ∧ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) ) → 𝑥 ≤ 𝑛 ) |
43 |
42
|
ralrimiva |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ∀ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) |
44 |
23
|
cnveqd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ◡ 𝑎 = ◡ 𝐴 ) |
45 |
44
|
imaeq1d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) = ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) ) |
46 |
45
|
raleqdv |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ( ∀ 𝑥 ∈ ( ◡ 𝑎 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ↔ ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) |
47 |
43 46
|
mpbid |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) ∧ ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) → ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) |
48 |
47
|
ex |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ ( 𝑛 ∈ ℕ0 ∧ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ) ) → ( ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) → ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) |
49 |
48
|
expr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) → ( ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) → ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) ) |
50 |
49
|
rexlimdv |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑛 ∈ ℕ0 ) → ( ∃ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) → ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) |
51 |
50
|
reximdva |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( ∃ 𝑛 ∈ ℕ0 ∃ 𝑎 ∈ ( ( 𝑆 ∪ { 0 } ) ↑m ℕ0 ) ( ( 𝑎 “ ( ℤ≥ ‘ ( 𝑛 + 1 ) ) ) = { 0 } ∧ 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝑎 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) → ∃ 𝑛 ∈ ℕ0 ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) |
52 |
3 51
|
mpd |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ∃ 𝑛 ∈ ℕ0 ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) |
53 |
|
ssrexv |
⊢ ( ℕ0 ⊆ ℤ → ( ∃ 𝑛 ∈ ℕ0 ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 → ∃ 𝑛 ∈ ℤ ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) |
54 |
29 52 53
|
mpsyl |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ∃ 𝑛 ∈ ℤ ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) |
55 |
28 54
|
jca |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( 𝐴 : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ∧ ∃ 𝑛 ∈ ℤ ∀ 𝑥 ∈ ( ◡ 𝐴 “ ( ℂ ∖ { 0 } ) ) 𝑥 ≤ 𝑛 ) ) |