Step |
Hyp |
Ref |
Expression |
0 |
|
cplylt |
⊢ Poly< |
1 |
|
vs |
⊢ 𝑠 |
2 |
|
cc |
⊢ ℂ |
3 |
2
|
cpw |
⊢ 𝒫 ℂ |
4 |
|
vx |
⊢ 𝑥 |
5 |
|
cn0 |
⊢ ℕ0 |
6 |
|
vp |
⊢ 𝑝 |
7 |
|
cply |
⊢ Poly |
8 |
1
|
cv |
⊢ 𝑠 |
9 |
8 7
|
cfv |
⊢ ( Poly ‘ 𝑠 ) |
10 |
6
|
cv |
⊢ 𝑝 |
11 |
|
c0p |
⊢ 0𝑝 |
12 |
10 11
|
wceq |
⊢ 𝑝 = 0𝑝 |
13 |
|
cdgr |
⊢ deg |
14 |
10 13
|
cfv |
⊢ ( deg ‘ 𝑝 ) |
15 |
|
clt |
⊢ < |
16 |
4
|
cv |
⊢ 𝑥 |
17 |
14 16 15
|
wbr |
⊢ ( deg ‘ 𝑝 ) < 𝑥 |
18 |
12 17
|
wo |
⊢ ( 𝑝 = 0𝑝 ∨ ( deg ‘ 𝑝 ) < 𝑥 ) |
19 |
18 6 9
|
crab |
⊢ { 𝑝 ∈ ( Poly ‘ 𝑠 ) ∣ ( 𝑝 = 0𝑝 ∨ ( deg ‘ 𝑝 ) < 𝑥 ) } |
20 |
1 4 3 5 19
|
cmpo |
⊢ ( 𝑠 ∈ 𝒫 ℂ , 𝑥 ∈ ℕ0 ↦ { 𝑝 ∈ ( Poly ‘ 𝑠 ) ∣ ( 𝑝 = 0𝑝 ∨ ( deg ‘ 𝑝 ) < 𝑥 ) } ) |
21 |
0 20
|
wceq |
⊢ Poly< = ( 𝑠 ∈ 𝒫 ℂ , 𝑥 ∈ ℕ0 ↦ { 𝑝 ∈ ( Poly ‘ 𝑠 ) ∣ ( 𝑝 = 0𝑝 ∨ ( deg ‘ 𝑝 ) < 𝑥 ) } ) |