Step |
Hyp |
Ref |
Expression |
1 |
|
mplsca.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mplsca.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
3 |
|
mplsca.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 ) |
4 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 ) |
5 |
4 2 3
|
psrsca |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
6 |
|
fvex |
⊢ ( Base ‘ 𝑃 ) ∈ V |
7 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
8 |
1 4 7
|
mplval2 |
⊢ 𝑃 = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ 𝑃 ) ) |
9 |
|
eqid |
⊢ ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) |
10 |
8 9
|
resssca |
⊢ ( ( Base ‘ 𝑃 ) ∈ V → ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Scalar ‘ 𝑃 ) ) |
11 |
6 10
|
ax-mp |
⊢ ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Scalar ‘ 𝑃 ) |
12 |
5 11
|
eqtrdi |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑃 ) ) |