Step |
Hyp |
Ref |
Expression |
1 |
|
pwspjmhm.y |
⊢ 𝑌 = ( 𝑅 ↑s 𝐼 ) |
2 |
|
pwspjmhm.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
eqid |
⊢ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) |
4 |
|
eqid |
⊢ ( Base ‘ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) = ( Base ‘ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) |
5 |
|
simp2 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → 𝐼 ∈ 𝑉 ) |
6 |
|
fvexd |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( Scalar ‘ 𝑅 ) ∈ V ) |
7 |
|
fconst6g |
⊢ ( 𝑅 ∈ Mnd → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ Mnd ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ Mnd ) |
9 |
|
simp3 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → 𝐴 ∈ 𝐼 ) |
10 |
3 4 5 6 8 9
|
prdspjmhm |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) ↦ ( 𝑥 ‘ 𝐴 ) ) ∈ ( ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) MndHom ( ( 𝐼 × { 𝑅 } ) ‘ 𝐴 ) ) ) |
11 |
|
eqid |
⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 ) |
12 |
1 11
|
pwsval |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ) → 𝑌 = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) |
13 |
12
|
3adant3 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → 𝑌 = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) |
14 |
13
|
fveq2d |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( Base ‘ 𝑌 ) = ( Base ‘ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
15 |
2 14
|
eqtrid |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → 𝐵 = ( Base ‘ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
16 |
15
|
mpteq1d |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 ‘ 𝐴 ) ) = ( 𝑥 ∈ ( Base ‘ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ) ↦ ( 𝑥 ‘ 𝐴 ) ) ) |
17 |
|
fvconst2g |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐴 ∈ 𝐼 ) → ( ( 𝐼 × { 𝑅 } ) ‘ 𝐴 ) = 𝑅 ) |
18 |
17
|
3adant2 |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( ( 𝐼 × { 𝑅 } ) ‘ 𝐴 ) = 𝑅 ) |
19 |
18
|
eqcomd |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → 𝑅 = ( ( 𝐼 × { 𝑅 } ) ‘ 𝐴 ) ) |
20 |
13 19
|
oveq12d |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( 𝑌 MndHom 𝑅 ) = ( ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) MndHom ( ( 𝐼 × { 𝑅 } ) ‘ 𝐴 ) ) ) |
21 |
10 16 20
|
3eltr4d |
⊢ ( ( 𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼 ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 ‘ 𝐴 ) ) ∈ ( 𝑌 MndHom 𝑅 ) ) |