Step |
Hyp |
Ref |
Expression |
1 |
|
lvecpropd.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
lvecpropd.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
lvecpropd.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
4 |
|
lvecpropd.4 |
⊢ ( 𝜑 → 𝐹 = ( Scalar ‘ 𝐾 ) ) |
5 |
|
lvecpropd.5 |
⊢ ( 𝜑 → 𝐹 = ( Scalar ‘ 𝐿 ) ) |
6 |
|
lvecpropd.6 |
⊢ 𝑃 = ( Base ‘ 𝐹 ) |
7 |
|
lvecpropd.7 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐿 ) 𝑦 ) ) |
8 |
1 2 3 4 5 6 7
|
lmodpropd |
⊢ ( 𝜑 → ( 𝐾 ∈ LMod ↔ 𝐿 ∈ LMod ) ) |
9 |
4 5
|
eqtr3d |
⊢ ( 𝜑 → ( Scalar ‘ 𝐾 ) = ( Scalar ‘ 𝐿 ) ) |
10 |
9
|
eleq1d |
⊢ ( 𝜑 → ( ( Scalar ‘ 𝐾 ) ∈ DivRing ↔ ( Scalar ‘ 𝐿 ) ∈ DivRing ) ) |
11 |
8 10
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝐾 ∈ LMod ∧ ( Scalar ‘ 𝐾 ) ∈ DivRing ) ↔ ( 𝐿 ∈ LMod ∧ ( Scalar ‘ 𝐿 ) ∈ DivRing ) ) ) |
12 |
|
eqid |
⊢ ( Scalar ‘ 𝐾 ) = ( Scalar ‘ 𝐾 ) |
13 |
12
|
islvec |
⊢ ( 𝐾 ∈ LVec ↔ ( 𝐾 ∈ LMod ∧ ( Scalar ‘ 𝐾 ) ∈ DivRing ) ) |
14 |
|
eqid |
⊢ ( Scalar ‘ 𝐿 ) = ( Scalar ‘ 𝐿 ) |
15 |
14
|
islvec |
⊢ ( 𝐿 ∈ LVec ↔ ( 𝐿 ∈ LMod ∧ ( Scalar ‘ 𝐿 ) ∈ DivRing ) ) |
16 |
11 13 15
|
3bitr4g |
⊢ ( 𝜑 → ( 𝐾 ∈ LVec ↔ 𝐿 ∈ LVec ) ) |