Metamath Proof Explorer


Definition df-plylt

Description: Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014)

Ref Expression
Assertion df-plylt Poly< = ( 𝑠 ∈ 𝒫 ℂ , 𝑥 ∈ ℕ0 ↦ { 𝑝 ∈ ( Poly ‘ 𝑠 ) ∣ ( 𝑝 = 0𝑝 ∨ ( deg ‘ 𝑝 ) < 𝑥 ) } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cplylt Poly<
1 vs 𝑠
2 cc
3 2 cpw 𝒫 ℂ
4 vx 𝑥
5 cn0 0
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 ) } )