| Step |
Hyp |
Ref |
Expression |
| 1 |
|
plycj.2 |
⊢ 𝐺 = ( ( ∗ ∘ 𝐹 ) ∘ ∗ ) |
| 2 |
|
plycj.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( ∗ ‘ 𝑥 ) ∈ 𝑆 ) |
| 3 |
|
plycj.4 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
| 4 |
|
eqid |
⊢ ( deg ‘ 𝐹 ) = ( deg ‘ 𝐹 ) |
| 5 |
|
eqid |
⊢ ( coeff ‘ 𝐹 ) = ( coeff ‘ 𝐹 ) |
| 6 |
4 1 5
|
plycjlem |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ( ( ( ∗ ∘ ( coeff ‘ 𝐹 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
| 7 |
3 6
|
syl |
⊢ ( 𝜑 → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ( ( ( ∗ ∘ ( coeff ‘ 𝐹 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
| 8 |
|
plybss |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑆 ⊆ ℂ ) |
| 9 |
3 8
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
| 10 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
| 11 |
10
|
snssd |
⊢ ( 𝜑 → { 0 } ⊆ ℂ ) |
| 12 |
9 11
|
unssd |
⊢ ( 𝜑 → ( 𝑆 ∪ { 0 } ) ⊆ ℂ ) |
| 13 |
|
dgrcl |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( deg ‘ 𝐹 ) ∈ ℕ0 ) |
| 14 |
3 13
|
syl |
⊢ ( 𝜑 → ( deg ‘ 𝐹 ) ∈ ℕ0 ) |
| 15 |
5
|
coef |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
| 16 |
3 15
|
syl |
⊢ ( 𝜑 → ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ) |
| 17 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) → 𝑘 ∈ ℕ0 ) |
| 18 |
|
fvco3 |
⊢ ( ( ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ∗ ∘ ( coeff ‘ 𝐹 ) ) ‘ 𝑘 ) = ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ) |
| 19 |
16 17 18
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ) → ( ( ∗ ∘ ( coeff ‘ 𝐹 ) ) ‘ 𝑘 ) = ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ) |
| 20 |
|
ffvelcdm |
⊢ ( ( ( coeff ‘ 𝐹 ) : ℕ0 ⟶ ( 𝑆 ∪ { 0 } ) ∧ 𝑘 ∈ ℕ0 ) → ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) ) |
| 21 |
16 17 20
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ) → ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) ) |
| 22 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( ∗ ‘ 𝑥 ) ∈ 𝑆 ) |
| 23 |
|
fveq2 |
⊢ ( 𝑥 = ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) → ( ∗ ‘ 𝑥 ) = ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ) |
| 24 |
23
|
eleq1d |
⊢ ( 𝑥 = ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) → ( ( ∗ ‘ 𝑥 ) ∈ 𝑆 ↔ ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ 𝑆 ) ) |
| 25 |
24
|
rspccv |
⊢ ( ∀ 𝑥 ∈ 𝑆 ( ∗ ‘ 𝑥 ) ∈ 𝑆 → ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ 𝑆 → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ 𝑆 ) ) |
| 26 |
22 25
|
syl |
⊢ ( 𝜑 → ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ 𝑆 → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ 𝑆 ) ) |
| 27 |
|
elsni |
⊢ ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } → ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) = 0 ) |
| 28 |
27
|
fveq2d |
⊢ ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) = ( ∗ ‘ 0 ) ) |
| 29 |
|
cj0 |
⊢ ( ∗ ‘ 0 ) = 0 |
| 30 |
28 29
|
eqtrdi |
⊢ ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) = 0 ) |
| 31 |
|
fvex |
⊢ ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ V |
| 32 |
31
|
elsn |
⊢ ( ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ { 0 } ↔ ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) = 0 ) |
| 33 |
30 32
|
sylibr |
⊢ ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ { 0 } ) |
| 34 |
33
|
a1i |
⊢ ( 𝜑 → ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ { 0 } ) ) |
| 35 |
26 34
|
orim12d |
⊢ ( 𝜑 → ( ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ 𝑆 ∨ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } ) → ( ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ 𝑆 ∨ ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ { 0 } ) ) ) |
| 36 |
|
elun |
⊢ ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) ↔ ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ 𝑆 ∨ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ { 0 } ) ) |
| 37 |
|
elun |
⊢ ( ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ ( 𝑆 ∪ { 0 } ) ↔ ( ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ 𝑆 ∨ ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ { 0 } ) ) |
| 38 |
35 36 37
|
3imtr4g |
⊢ ( 𝜑 → ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ ( 𝑆 ∪ { 0 } ) ) ) |
| 39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ) → ( ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ ( 𝑆 ∪ { 0 } ) ) ) |
| 40 |
21 39
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ) → ( ∗ ‘ ( ( coeff ‘ 𝐹 ) ‘ 𝑘 ) ) ∈ ( 𝑆 ∪ { 0 } ) ) |
| 41 |
19 40
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ) → ( ( ∗ ∘ ( coeff ‘ 𝐹 ) ) ‘ 𝑘 ) ∈ ( 𝑆 ∪ { 0 } ) ) |
| 42 |
12 14 41
|
elplyd |
⊢ ( 𝜑 → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... ( deg ‘ 𝐹 ) ) ( ( ( ∗ ∘ ( coeff ‘ 𝐹 ) ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ∈ ( Poly ‘ ( 𝑆 ∪ { 0 } ) ) ) |
| 43 |
7 42
|
eqeltrd |
⊢ ( 𝜑 → 𝐺 ∈ ( Poly ‘ ( 𝑆 ∪ { 0 } ) ) ) |
| 44 |
|
plyun0 |
⊢ ( Poly ‘ ( 𝑆 ∪ { 0 } ) ) = ( Poly ‘ 𝑆 ) |
| 45 |
43 44
|
eleqtrdi |
⊢ ( 𝜑 → 𝐺 ∈ ( Poly ‘ 𝑆 ) ) |