Step |
Hyp |
Ref |
Expression |
1 |
|
pwslmod.y |
⊢ 𝑌 = ( 𝑅 ↑s 𝐼 ) |
2 |
|
eqid |
⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 ) |
3 |
1 2
|
pwsval |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) → 𝑌 = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) |
4 |
|
eqid |
⊢ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) |
5 |
2
|
lmodring |
⊢ ( 𝑅 ∈ LMod → ( Scalar ‘ 𝑅 ) ∈ Ring ) |
6 |
5
|
adantr |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) → ( Scalar ‘ 𝑅 ) ∈ Ring ) |
7 |
|
simpr |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) → 𝐼 ∈ 𝑉 ) |
8 |
|
fconst6g |
⊢ ( 𝑅 ∈ LMod → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ LMod ) |
9 |
8
|
adantr |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ LMod ) |
10 |
|
fvconst2g |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) = 𝑅 ) |
11 |
10
|
adantlr |
⊢ ( ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) = 𝑅 ) |
12 |
11
|
fveq2d |
⊢ ( ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑥 ∈ 𝐼 ) → ( Scalar ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) = ( Scalar ‘ 𝑅 ) ) |
13 |
4 6 7 9 12
|
prdslmodd |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) → ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ∈ LMod ) |
14 |
3 13
|
eqeltrd |
⊢ ( ( 𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉 ) → 𝑌 ∈ LMod ) |