Step |
Hyp |
Ref |
Expression |
1 |
|
mplgrp.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
drngring |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring ) |
3 |
1
|
mpllmod |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ LMod ) |
4 |
2 3
|
sylan2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing ) → 𝑃 ∈ LMod ) |
5 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing ) → 𝐼 ∈ 𝑉 ) |
6 |
|
simpr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing ) → 𝑅 ∈ DivRing ) |
7 |
1 5 6
|
mplsca |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing ) → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
8 |
7 6
|
eqeltrrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing ) → ( Scalar ‘ 𝑃 ) ∈ DivRing ) |
9 |
|
eqid |
⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 ) |
10 |
9
|
islvec |
⊢ ( 𝑃 ∈ LVec ↔ ( 𝑃 ∈ LMod ∧ ( Scalar ‘ 𝑃 ) ∈ DivRing ) ) |
11 |
4 8 10
|
sylanbrc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing ) → 𝑃 ∈ LVec ) |