Step |
Hyp |
Ref |
Expression |
1 |
|
pwssca.y |
⊢ 𝑌 = ( 𝑅 ↑s 𝐼 ) |
2 |
|
pwssca.s |
⊢ 𝑆 = ( Scalar ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( 𝑆 Xs ( 𝐼 × { 𝑅 } ) ) = ( 𝑆 Xs ( 𝐼 × { 𝑅 } ) ) |
4 |
2
|
fvexi |
⊢ 𝑆 ∈ V |
5 |
4
|
a1i |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝑆 ∈ V ) |
6 |
|
simpr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐼 ∈ 𝑊 ) |
7 |
|
snex |
⊢ { 𝑅 } ∈ V |
8 |
|
xpexg |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ { 𝑅 } ∈ V ) → ( 𝐼 × { 𝑅 } ) ∈ V ) |
9 |
6 7 8
|
sylancl |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝐼 × { 𝑅 } ) ∈ V ) |
10 |
3 5 9
|
prdssca |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝑆 = ( Scalar ‘ ( 𝑆 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
11 |
1 2
|
pwsval |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝑌 = ( 𝑆 Xs ( 𝐼 × { 𝑅 } ) ) ) |
12 |
11
|
fveq2d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( Scalar ‘ 𝑌 ) = ( Scalar ‘ ( 𝑆 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
13 |
10 12
|
eqtr4d |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝑆 = ( Scalar ‘ 𝑌 ) ) |