Step |
Hyp |
Ref |
Expression |
1 |
|
islidl.s |
⊢ 𝑈 = ( LIdeal ‘ 𝑅 ) |
2 |
|
islidl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
islidl.p |
⊢ + = ( +g ‘ 𝑅 ) |
4 |
|
islidl.t |
⊢ · = ( .r ‘ 𝑅 ) |
5 |
|
rlmsca2 |
⊢ ( I ‘ 𝑅 ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) |
6 |
|
baseid |
⊢ Base = Slot ( Base ‘ ndx ) |
7 |
6 2
|
strfvi |
⊢ 𝐵 = ( Base ‘ ( I ‘ 𝑅 ) ) |
8 |
|
rlmbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
9 |
2 8
|
eqtri |
⊢ 𝐵 = ( Base ‘ ( ringLMod ‘ 𝑅 ) ) |
10 |
|
rlmplusg |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ ( ringLMod ‘ 𝑅 ) ) |
11 |
3 10
|
eqtri |
⊢ + = ( +g ‘ ( ringLMod ‘ 𝑅 ) ) |
12 |
|
rlmvsca |
⊢ ( .r ‘ 𝑅 ) = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
13 |
4 12
|
eqtri |
⊢ · = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
14 |
|
lidlval |
⊢ ( LIdeal ‘ 𝑅 ) = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) ) |
15 |
1 14
|
eqtri |
⊢ 𝑈 = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) ) |
16 |
5 7 9 11 13 15
|
islss |
⊢ ( 𝐼 ∈ 𝑈 ↔ ( 𝐼 ⊆ 𝐵 ∧ 𝐼 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ 𝐼 ∀ 𝑏 ∈ 𝐼 ( ( 𝑥 · 𝑎 ) + 𝑏 ) ∈ 𝐼 ) ) |