Metamath Proof Explorer


Theorem uc1pdeg

Description: Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses uc1pdeg.d 𝐷 = ( deg1𝑅 )
uc1pdeg.c 𝐶 = ( Unic1p𝑅 )
Assertion uc1pdeg ( ( 𝑅 ∈ Ring ∧ 𝐹𝐶 ) → ( 𝐷𝐹 ) ∈ ℕ0 )

Proof

Step Hyp Ref Expression
1 uc1pdeg.d 𝐷 = ( deg1𝑅 )
2 uc1pdeg.c 𝐶 = ( Unic1p𝑅 )
3 simpl ( ( 𝑅 ∈ Ring ∧ 𝐹𝐶 ) → 𝑅 ∈ Ring )
4 eqid ( Poly1𝑅 ) = ( Poly1𝑅 )
5 eqid ( Base ‘ ( Poly1𝑅 ) ) = ( Base ‘ ( Poly1𝑅 ) )
6 4 5 2 uc1pcl ( 𝐹𝐶𝐹 ∈ ( Base ‘ ( Poly1𝑅 ) ) )
7 6 adantl ( ( 𝑅 ∈ Ring ∧ 𝐹𝐶 ) → 𝐹 ∈ ( Base ‘ ( Poly1𝑅 ) ) )
8 eqid ( 0g ‘ ( Poly1𝑅 ) ) = ( 0g ‘ ( Poly1𝑅 ) )
9 4 8 2 uc1pn0 ( 𝐹𝐶𝐹 ≠ ( 0g ‘ ( Poly1𝑅 ) ) )
10 9 adantl ( ( 𝑅 ∈ Ring ∧ 𝐹𝐶 ) → 𝐹 ≠ ( 0g ‘ ( Poly1𝑅 ) ) )
11 1 4 8 5 deg1nn0cl ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( Base ‘ ( Poly1𝑅 ) ) ∧ 𝐹 ≠ ( 0g ‘ ( Poly1𝑅 ) ) ) → ( 𝐷𝐹 ) ∈ ℕ0 )
12 3 7 10 11 syl3anc ( ( 𝑅 ∈ Ring ∧ 𝐹𝐶 ) → ( 𝐷𝐹 ) ∈ ℕ0 )