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Metamath Proof Explorer

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 )