Theorem mon1pcl

Description: Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	uc1pcl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	uc1pcl.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	mon1pcl.m	⊢ 𝑀 = ( Monic1p ‘ 𝑅 )
Assertion	mon1pcl	⊢ ( 𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		uc1pcl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		uc1pcl.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
3		mon1pcl.m	⊢ 𝑀 = ( Monic1p ‘ 𝑅 )
4		eqid	⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 )
5		eqid	⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 )
6		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
7	1 2 4 5 3 6	ismon1p	⊢ ( 𝐹 ∈ 𝑀 ↔ ( 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ ( 0g ‘ 𝑃 ) ∧ ( ( coe1 ‘ 𝐹 ) ‘ ( ( deg1 ‘ 𝑅 ) ‘ 𝐹 ) ) = ( 1r ‘ 𝑅 ) ) )
8	7	simp1bi	⊢ ( 𝐹 ∈ 𝑀 → 𝐹 ∈ 𝐵 )