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 |
|
r1pdeglt.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
6 |
|
simp2 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐹 ∈ 𝐵 ) |
7 |
2 3 4
|
uc1pcl |
⊢ ( 𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵 ) |
8 |
7
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐺 ∈ 𝐵 ) |
9 |
|
eqid |
⊢ ( quot1p ‘ 𝑅 ) = ( quot1p ‘ 𝑅 ) |
10 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
11 |
|
eqid |
⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 ) |
12 |
1 2 3 9 10 11
|
r1pval |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) |
13 |
6 8 12
|
syl2anc |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) |
14 |
13
|
fveq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) = ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) ) |
15 |
|
eqid |
⊢ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) |
16 |
9 2 3 5 11 10 4
|
q1peqb |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) ↔ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ) ) |
17 |
15 16
|
mpbiri |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) ) |
18 |
17
|
simprd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) |
19 |
14 18
|
eqbrtrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷 ‘ 𝐺 ) ) |