Step |
Hyp |
Ref |
Expression |
1 |
|
ply1vscl.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ply1vscl.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
ply1vscl.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
ply1vscl.s |
⊢ · = ( ·𝑠 ‘ 𝑃 ) |
5 |
|
ply1vscl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
6 |
|
ply1vscl.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐾 ) |
7 |
|
ply1vscl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
1 2
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
9 |
|
eqid |
⊢ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) = ( Scalar ‘ ( 1o mPoly 𝑅 ) ) |
10 |
|
eqid |
⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 ) |
11 |
1 10 4
|
ply1vsca |
⊢ · = ( ·𝑠 ‘ ( 1o mPoly 𝑅 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) |
13 |
|
1oex |
⊢ 1o ∈ V |
14 |
13
|
a1i |
⊢ ( 𝜑 → 1o ∈ V ) |
15 |
10 14 5
|
mpllmodd |
⊢ ( 𝜑 → ( 1o mPoly 𝑅 ) ∈ LMod ) |
16 |
10 14 5
|
mplsca |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) |
17 |
16
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) ) |
18 |
3 17
|
eqtrid |
⊢ ( 𝜑 → 𝐾 = ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) ) |
19 |
6 18
|
eleqtrd |
⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ ( Scalar ‘ ( 1o mPoly 𝑅 ) ) ) ) |
20 |
8 9 11 12 15 19 7
|
lmodvscld |
⊢ ( 𝜑 → ( 𝐶 · 𝑋 ) ∈ 𝐵 ) |