Theorem mon1pn0

Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	uc1pn0.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	uc1pn0.z	⊢ 0 = ( 0g ‘ 𝑃 )
	mon1pn0.m	⊢ 𝑀 = ( Monic1p ‘ 𝑅 )
Assertion	mon1pn0	⊢ ( 𝐹 ∈ 𝑀 → 𝐹 ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		uc1pn0.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		uc1pn0.z	⊢ 0 = ( 0g ‘ 𝑃 )
3		mon1pn0.m	⊢ 𝑀 = ( Monic1p ‘ 𝑅 )
4		eqid	⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
5		eqid	⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 )
6		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
7	1 4 2 5 3 6	ismon1p	⊢ ( 𝐹 ∈ 𝑀 ↔ ( 𝐹 ∈ ( Base ‘ 𝑃 ) ∧ 𝐹 ≠ 0 ∧ ( ( coe1 ‘ 𝐹 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝐹 ) ) = ( 1r ‘ 𝑅 ) ) )
8	7	simp2bi	⊢ ( 𝐹 ∈ 𝑀 → 𝐹 ≠ 0 )