Step |
Hyp |
Ref |
Expression |
1 |
|
ascl0.a |
⊢ 𝐴 = ( algSc ‘ 𝑊 ) |
2 |
|
ascl0.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
ascl0.l |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
4 |
|
ascl0.r |
⊢ ( 𝜑 → 𝑊 ∈ Ring ) |
5 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 ) |
7 |
|
eqid |
⊢ ( 1r ‘ 𝐹 ) = ( 1r ‘ 𝐹 ) |
8 |
6 7
|
ringidcl |
⊢ ( 𝐹 ∈ Ring → ( 1r ‘ 𝐹 ) ∈ ( Base ‘ 𝐹 ) ) |
9 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
10 |
|
eqid |
⊢ ( 1r ‘ 𝑊 ) = ( 1r ‘ 𝑊 ) |
11 |
1 2 6 9 10
|
asclval |
⊢ ( ( 1r ‘ 𝐹 ) ∈ ( Base ‘ 𝐹 ) → ( 𝐴 ‘ ( 1r ‘ 𝐹 ) ) = ( ( 1r ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
12 |
3 5 8 11
|
4syl |
⊢ ( 𝜑 → ( 𝐴 ‘ ( 1r ‘ 𝐹 ) ) = ( ( 1r ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
14 |
13 10
|
ringidcl |
⊢ ( 𝑊 ∈ Ring → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
15 |
4 14
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) |
16 |
13 2 9 7
|
lmodvs1 |
⊢ ( ( 𝑊 ∈ LMod ∧ ( 1r ‘ 𝑊 ) ∈ ( Base ‘ 𝑊 ) ) → ( ( 1r ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( 1r ‘ 𝑊 ) ) |
17 |
3 15 16
|
syl2anc |
⊢ ( 𝜑 → ( ( 1r ‘ 𝐹 ) ( ·𝑠 ‘ 𝑊 ) ( 1r ‘ 𝑊 ) ) = ( 1r ‘ 𝑊 ) ) |
18 |
12 17
|
eqtrd |
⊢ ( 𝜑 → ( 𝐴 ‘ ( 1r ‘ 𝐹 ) ) = ( 1r ‘ 𝑊 ) ) |