Step |
Hyp |
Ref |
Expression |
1 |
|
dflidl2.u |
⊢ 𝑈 = ( LIdeal ‘ 𝑅 ) |
2 |
|
dflidl2.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
dflidl2.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
1
|
lidlsubg |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ) |
5 |
1 2 3
|
lidlmcl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐼 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝐼 ) |
6 |
5
|
ralrimivva |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ) |
7 |
4 6
|
jca |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ) → ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ) ) |
8 |
1 2 3
|
dflidl2lem |
⊢ ( ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ) → 𝐼 ∈ 𝑈 ) |
9 |
8
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ) ) → 𝐼 ∈ 𝑈 ) |
10 |
7 9
|
impbida |
⊢ ( 𝑅 ∈ Ring → ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ∈ ( SubGrp ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐼 ( 𝑥 · 𝑦 ) ∈ 𝐼 ) ) ) |