Step |
Hyp |
Ref |
Expression |
1 |
|
ig1pval.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ig1pval.g |
⊢ 𝐺 = ( idlGen1p ‘ 𝑅 ) |
3 |
|
ig1pcl.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑃 ) |
4 |
|
ig1prsp.k |
⊢ 𝐾 = ( RSpan ‘ 𝑃 ) |
5 |
1 2 3
|
ig1pcl |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ) |
6 |
|
eqid |
⊢ ( ∥r ‘ 𝑃 ) = ( ∥r ‘ 𝑃 ) |
7 |
1 2 3 6
|
ig1pdvds |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝐼 ) ( ∥r ‘ 𝑃 ) 𝑥 ) |
8 |
7
|
3expa |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝐼 ) ( ∥r ‘ 𝑃 ) 𝑥 ) |
9 |
8
|
ralrimiva |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝐼 ) ( ∥r ‘ 𝑃 ) 𝑥 ) |
10 |
|
drngring |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring ) |
11 |
1
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
12 |
10 11
|
syl |
⊢ ( 𝑅 ∈ DivRing → 𝑃 ∈ Ring ) |
13 |
12
|
adantr |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → 𝑃 ∈ Ring ) |
14 |
|
simpr |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → 𝐼 ∈ 𝑈 ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
16 |
15 3
|
lidlss |
⊢ ( 𝐼 ∈ 𝑈 → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → 𝐼 ⊆ ( Base ‘ 𝑃 ) ) |
18 |
17 5
|
sseldd |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → ( 𝐺 ‘ 𝐼 ) ∈ ( Base ‘ 𝑃 ) ) |
19 |
15 3 4 6
|
lidldvgen |
⊢ ( ( 𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ ( 𝐺 ‘ 𝐼 ) ∈ ( Base ‘ 𝑃 ) ) → ( 𝐼 = ( 𝐾 ‘ { ( 𝐺 ‘ 𝐼 ) } ) ↔ ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝐼 ) ( ∥r ‘ 𝑃 ) 𝑥 ) ) ) |
20 |
13 14 18 19
|
syl3anc |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → ( 𝐼 = ( 𝐾 ‘ { ( 𝐺 ‘ 𝐼 ) } ) ↔ ( ( 𝐺 ‘ 𝐼 ) ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐺 ‘ 𝐼 ) ( ∥r ‘ 𝑃 ) 𝑥 ) ) ) |
21 |
5 9 20
|
mpbir2and |
⊢ ( ( 𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ) → 𝐼 = ( 𝐾 ‘ { ( 𝐺 ‘ 𝐼 ) } ) ) |