Step |
Hyp |
Ref |
Expression |
1 |
|
ssid |
⊢ ℂ ⊆ ℂ |
2 |
|
plyconst |
⊢ ( ( ℂ ⊆ ℂ ∧ 𝐴 ∈ ℂ ) → ( ℂ × { 𝐴 } ) ∈ ( Poly ‘ ℂ ) ) |
3 |
1 2
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( ℂ × { 𝐴 } ) ∈ ( Poly ‘ ℂ ) ) |
4 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
5 |
4
|
a1i |
⊢ ( 𝐴 ∈ ℂ → 0 ∈ ℕ0 ) |
6 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ( 0 ... 0 ) ) → 𝐴 ∈ ℂ ) |
7 |
|
fconstmpt |
⊢ ( ℂ × { 𝐴 } ) = ( 𝑧 ∈ ℂ ↦ 𝐴 ) |
8 |
|
0z |
⊢ 0 ∈ ℤ |
9 |
|
exp0 |
⊢ ( 𝑧 ∈ ℂ → ( 𝑧 ↑ 0 ) = 1 ) |
10 |
9
|
oveq2d |
⊢ ( 𝑧 ∈ ℂ → ( 𝐴 · ( 𝑧 ↑ 0 ) ) = ( 𝐴 · 1 ) ) |
11 |
|
mulid1 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 1 ) = 𝐴 ) |
12 |
10 11
|
sylan9eqr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( 𝐴 · ( 𝑧 ↑ 0 ) ) = 𝐴 ) |
13 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
14 |
12 13
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( 𝐴 · ( 𝑧 ↑ 0 ) ) ∈ ℂ ) |
15 |
|
oveq2 |
⊢ ( 𝑘 = 0 → ( 𝑧 ↑ 𝑘 ) = ( 𝑧 ↑ 0 ) ) |
16 |
15
|
oveq2d |
⊢ ( 𝑘 = 0 → ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 0 ) ) ) |
17 |
16
|
fsum1 |
⊢ ( ( 0 ∈ ℤ ∧ ( 𝐴 · ( 𝑧 ↑ 0 ) ) ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 0 ) ) ) |
18 |
8 14 17
|
sylancr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 0 ) ) ) |
19 |
18 12
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) = 𝐴 ) |
20 |
19
|
mpteq2dva |
⊢ ( 𝐴 ∈ ℂ → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ 𝐴 ) ) |
21 |
7 20
|
eqtr4id |
⊢ ( 𝐴 ∈ ℂ → ( ℂ × { 𝐴 } ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝐴 · ( 𝑧 ↑ 𝑘 ) ) ) ) |
22 |
3 5 6 21
|
dgrle |
⊢ ( 𝐴 ∈ ℂ → ( deg ‘ ( ℂ × { 𝐴 } ) ) ≤ 0 ) |
23 |
|
dgrcl |
⊢ ( ( ℂ × { 𝐴 } ) ∈ ( Poly ‘ ℂ ) → ( deg ‘ ( ℂ × { 𝐴 } ) ) ∈ ℕ0 ) |
24 |
|
nn0le0eq0 |
⊢ ( ( deg ‘ ( ℂ × { 𝐴 } ) ) ∈ ℕ0 → ( ( deg ‘ ( ℂ × { 𝐴 } ) ) ≤ 0 ↔ ( deg ‘ ( ℂ × { 𝐴 } ) ) = 0 ) ) |
25 |
3 23 24
|
3syl |
⊢ ( 𝐴 ∈ ℂ → ( ( deg ‘ ( ℂ × { 𝐴 } ) ) ≤ 0 ↔ ( deg ‘ ( ℂ × { 𝐴 } ) ) = 0 ) ) |
26 |
22 25
|
mpbid |
⊢ ( 𝐴 ∈ ℂ → ( deg ‘ ( ℂ × { 𝐴 } ) ) = 0 ) |