Metamath Proof Explorer


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 = ( rem1p𝑅 )
r1pval.p 𝑃 = ( Poly1𝑅 )
r1pval.b 𝐵 = ( Base ‘ 𝑃 )
r1pcl.c 𝐶 = ( Unic1p𝑅 )
Assertion r1pcl ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐹 𝐸 𝐺 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 r1pval.e 𝐸 = ( rem1p𝑅 )
2 r1pval.p 𝑃 = ( Poly1𝑅 )
3 r1pval.b 𝐵 = ( Base ‘ 𝑃 )
4 r1pcl.c 𝐶 = ( Unic1p𝑅 )
5 simp2 ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → 𝐹𝐵 )
6 2 3 4 uc1pcl ( 𝐺𝐶𝐺𝐵 )
7 6 3ad2ant3 ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → 𝐺𝐵 )
8 eqid ( quot1p𝑅 ) = ( quot1p𝑅 )
9 eqid ( .r𝑃 ) = ( .r𝑃 )
10 eqid ( -g𝑃 ) = ( -g𝑃 )
11 1 2 3 8 9 10 r1pval ( ( 𝐹𝐵𝐺𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) )
12 5 7 11 syl2anc ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) )
13 2 ply1ring ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
14 13 3ad2ant1 ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → 𝑃 ∈ Ring )
15 ringgrp ( 𝑃 ∈ Ring → 𝑃 ∈ Grp )
16 14 15 syl ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → 𝑃 ∈ Grp )
17 8 2 3 4 q1pcl ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ∈ 𝐵 )
18 3 9 ringcl ( ( 𝑃 ∈ Ring ∧ ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ∈ 𝐵𝐺𝐵 ) → ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ∈ 𝐵 )
19 14 17 7 18 syl3anc ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ∈ 𝐵 )
20 3 10 grpsubcl ( ( 𝑃 ∈ Grp ∧ 𝐹𝐵 ∧ ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ∈ 𝐵 ) → ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) ∈ 𝐵 )
21 16 5 19 20 syl3anc ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) ∈ 𝐵 )
22 12 21 eqeltrd ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐹 𝐸 𝐺 ) ∈ 𝐵 )