Step |
Hyp |
Ref |
Expression |
1 |
|
mplasclf.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mplasclf.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
mplasclf.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
mplasclf.a |
⊢ 𝐴 = ( algSc ‘ 𝑃 ) |
5 |
|
mplasclf.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
6 |
|
mplasclf.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
7 |
|
eqid |
⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 ) |
8 |
1
|
mplring |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ Ring ) |
9 |
1
|
mpllmod |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ LMod ) |
10 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) |
11 |
4 7 8 9 10 2
|
asclf |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring ) → 𝐴 : ( Base ‘ ( Scalar ‘ 𝑃 ) ) ⟶ 𝐵 ) |
12 |
5 6 11
|
syl2anc |
⊢ ( 𝜑 → 𝐴 : ( Base ‘ ( Scalar ‘ 𝑃 ) ) ⟶ 𝐵 ) |
13 |
1 5 6
|
mplsca |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
14 |
13
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) |
15 |
3 14
|
syl5eq |
⊢ ( 𝜑 → 𝐾 = ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) |
16 |
15
|
feq2d |
⊢ ( 𝜑 → ( 𝐴 : 𝐾 ⟶ 𝐵 ↔ 𝐴 : ( Base ‘ ( Scalar ‘ 𝑃 ) ) ⟶ 𝐵 ) ) |
17 |
12 16
|
mpbird |
⊢ ( 𝜑 → 𝐴 : 𝐾 ⟶ 𝐵 ) |