Step |
Hyp |
Ref |
Expression |
1 |
|
frlmvscafval.y |
⊢ 𝑌 = ( 𝑅 freeLMod 𝐼 ) |
2 |
|
frlmvscafval.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
frlmvscafval.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
frlmvscafval.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
frlmvscafval.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐾 ) |
6 |
|
frlmvscafval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
7 |
|
frlmvscafval.v |
⊢ ∙ = ( ·𝑠 ‘ 𝑌 ) |
8 |
|
frlmvscafval.t |
⊢ · = ( .r ‘ 𝑅 ) |
9 |
1 2
|
frlmrcl |
⊢ ( 𝑋 ∈ 𝐵 → 𝑅 ∈ V ) |
10 |
6 9
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
11 |
1 2
|
frlmpws |
⊢ ( ( 𝑅 ∈ V ∧ 𝐼 ∈ 𝑊 ) → 𝑌 = ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) |
12 |
10 4 11
|
syl2anc |
⊢ ( 𝜑 → 𝑌 = ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) |
13 |
12
|
fveq2d |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑌 ) = ( ·𝑠 ‘ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) ) |
14 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
15 |
|
eqid |
⊢ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) = ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) |
16 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) = ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) |
17 |
15 16
|
ressvsca |
⊢ ( 𝐵 ∈ V → ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) = ( ·𝑠 ‘ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) ) |
18 |
14 17
|
ax-mp |
⊢ ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) = ( ·𝑠 ‘ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) |
19 |
13 7 18
|
3eqtr4g |
⊢ ( 𝜑 → ∙ = ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) ) |
20 |
19
|
oveqd |
⊢ ( 𝜑 → ( 𝐴 ∙ 𝑋 ) = ( 𝐴 ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) 𝑋 ) ) |
21 |
|
eqid |
⊢ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) = ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) |
22 |
|
eqid |
⊢ ( Base ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) = ( Base ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) |
23 |
|
rlmvsca |
⊢ ( .r ‘ 𝑅 ) = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
24 |
8 23
|
eqtri |
⊢ · = ( ·𝑠 ‘ ( ringLMod ‘ 𝑅 ) ) |
25 |
|
eqid |
⊢ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) |
26 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
27 |
|
fvexd |
⊢ ( 𝜑 → ( ringLMod ‘ 𝑅 ) ∈ V ) |
28 |
|
rlmsca |
⊢ ( 𝑅 ∈ V → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
29 |
10 28
|
syl |
⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) |
30 |
29
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
31 |
3 30
|
syl5eq |
⊢ ( 𝜑 → 𝐾 = ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
32 |
5 31
|
eleqtrd |
⊢ ( 𝜑 → 𝐴 ∈ ( Base ‘ ( Scalar ‘ ( ringLMod ‘ 𝑅 ) ) ) ) |
33 |
12
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝑌 ) = ( Base ‘ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) ) |
34 |
2 33
|
syl5eq |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) ) |
35 |
15 22
|
ressbasss |
⊢ ( Base ‘ ( ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ↾s 𝐵 ) ) ⊆ ( Base ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) |
36 |
34 35
|
eqsstrdi |
⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) ) |
37 |
36 6
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) ) |
38 |
21 22 24 16 25 26 27 4 32 37
|
pwsvscafval |
⊢ ( 𝜑 → ( 𝐴 ( ·𝑠 ‘ ( ( ringLMod ‘ 𝑅 ) ↑s 𝐼 ) ) 𝑋 ) = ( ( 𝐼 × { 𝐴 } ) ∘f · 𝑋 ) ) |
39 |
20 38
|
eqtrd |
⊢ ( 𝜑 → ( 𝐴 ∙ 𝑋 ) = ( ( 𝐼 × { 𝐴 } ) ∘f · 𝑋 ) ) |