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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 = ( 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 )