Theorem q1pcl

Description: Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	q1pcl.q	⊢ 𝑄 = ( quot1p ‘ 𝑅 )
	q1pcl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	q1pcl.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	q1pcl.c	⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
Assertion	q1pcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝑄 𝐺 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		q1pcl.q	⊢ 𝑄 = ( quot1p ‘ 𝑅 )
2		q1pcl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3		q1pcl.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		q1pcl.c	⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
5		eqid	⊢ ( 𝐹 𝑄 𝐺 ) = ( 𝐹 𝑄 𝐺 )
6		eqid	⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 )
7		eqid	⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 )
8		eqid	⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
9	1 2 3 6 7 8 4	q1peqb	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( ( 𝐹 𝑄 𝐺 ) ∈ 𝐵 ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 𝑄 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ) ↔ ( 𝐹 𝑄 𝐺 ) = ( 𝐹 𝑄 𝐺 ) ) )
10	5 9	mpbiri	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( 𝐹 𝑄 𝐺 ) ∈ 𝐵 ∧ ( ( deg1 ‘ 𝑅 ) ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 𝑄 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( ( deg1 ‘ 𝑅 ) ‘ 𝐺 ) ) )
11	10	simpld	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝑄 𝐺 ) ∈ 𝐵 )