Step |
Hyp |
Ref |
Expression |
1 |
|
pwscmn.y |
⊢ 𝑌 = ( 𝑅 ↑s 𝐼 ) |
2 |
|
eqid |
⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 ) |
3 |
1 2
|
pwsval |
⊢ ( ( 𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉 ) → 𝑌 = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) |
4 |
|
eqid |
⊢ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) |
5 |
|
simpr |
⊢ ( ( 𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉 ) → 𝐼 ∈ 𝑉 ) |
6 |
|
fvexd |
⊢ ( ( 𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉 ) → ( Scalar ‘ 𝑅 ) ∈ V ) |
7 |
|
fconst6g |
⊢ ( 𝑅 ∈ CMnd → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ CMnd ) |
8 |
7
|
adantr |
⊢ ( ( 𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉 ) → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ CMnd ) |
9 |
4 5 6 8
|
prdscmnd |
⊢ ( ( 𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉 ) → ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ∈ CMnd ) |
10 |
3 9
|
eqeltrd |
⊢ ( ( 𝑅 ∈ CMnd ∧ 𝐼 ∈ 𝑉 ) → 𝑌 ∈ CMnd ) |