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 ) |