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 ) |