Step |
Hyp |
Ref |
Expression |
1 |
|
mplvsca.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
2 |
|
mplvsca.n |
⊢ ∙ = ( ·𝑠 ‘ 𝑃 ) |
3 |
|
mplvsca.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
mplvsca.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
5 |
|
mplvsca.m |
⊢ · = ( .r ‘ 𝑅 ) |
6 |
|
mplvsca.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
7 |
|
mplvsca.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐾 ) |
8 |
|
mplvsca.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
9 |
|
eqid |
⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 ) |
10 |
1 9 2
|
mplvsca2 |
⊢ ∙ = ( ·𝑠 ‘ ( 𝐼 mPwSer 𝑅 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) |
12 |
1 9 4 11
|
mplbasss |
⊢ 𝐵 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) |
13 |
12 8
|
sselid |
⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) ) |
14 |
9 10 3 11 5 6 7 13
|
psrvsca |
⊢ ( 𝜑 → ( 𝑋 ∙ 𝐹 ) = ( ( 𝐷 × { 𝑋 } ) ∘f · 𝐹 ) ) |