Theorem r1pdeglt

Description: The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	r1pval.e	⊢ 𝐸 = ( rem1p ‘ 𝑅 )
	r1pval.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	r1pval.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	r1pcl.c	⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
	r1pdeglt.d	⊢ 𝐷 = ( deg1 ‘ 𝑅 )
Assertion	r1pdeglt	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷 ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		r1pval.e	⊢ 𝐸 = ( rem1p ‘ 𝑅 )
2		r1pval.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3		r1pval.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		r1pcl.c	⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
5		r1pdeglt.d	⊢ 𝐷 = ( deg1 ‘ 𝑅 )
6		simp2	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐹 ∈ 𝐵 )
7	2 3 4	uc1pcl	⊢ ( 𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵 )
8	7	3ad2ant3	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐺 ∈ 𝐵 )
9		eqid	⊢ ( quot1p ‘ 𝑅 ) = ( quot1p ‘ 𝑅 )
10		eqid	⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
11		eqid	⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 )
12	1 2 3 9 10 11	r1pval	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) )
13	6 8 12	syl2anc	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) )
14	13	fveq2d	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) = ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) )
15		eqid	⊢ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 )
16	9 2 3 5 11 10 4	q1peqb	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) ↔ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ) )
17	15 16	mpbiri	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) )
18	17	simprd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) )
19	14 18	eqbrtrd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷 ‘ 𝐺 ) )