Theorem r1pcl

Description: Closure of remainder following division by a unitic polynomial. (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 )
Assertion	r1pcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝐸 𝐺 ) ∈ 𝐵 )

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		simp2	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐹 ∈ 𝐵 )
6	2 3 4	uc1pcl	⊢ ( 𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵 )
7	6	3ad2ant3	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐺 ∈ 𝐵 )
8		eqid	⊢ ( quot1p ‘ 𝑅 ) = ( quot1p ‘ 𝑅 )
9		eqid	⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
10		eqid	⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 )
11	1 2 3 8 9 10	r1pval	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) )
12	5 7 11	syl2anc	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) )
13	2	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
14	13	3ad2ant1	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝑃 ∈ Ring )
15		ringgrp	⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ Grp )
16	14 15	syl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝑃 ∈ Grp )
17	8 2 3 4	q1pcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 )
18	3 9	ringcl	⊢ ( ( 𝑃 ∈ Ring ∧ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ∈ 𝐵 )
19	14 17 7 18	syl3anc	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ∈ 𝐵 )
20	3 10	grpsubcl	⊢ ( ( 𝑃 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ∈ 𝐵 ) → ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ∈ 𝐵 )
21	16 5 19 20	syl3anc	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ∈ 𝐵 )
22	12 21	eqeltrd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝐸 𝐺 ) ∈ 𝐵 )