Step |
Hyp |
Ref |
Expression |
1 |
|
lkrss.r |
⊢ 𝑅 = ( Scalar ‘ 𝑊 ) |
2 |
|
lkrss.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
3 |
|
lkrss.f |
⊢ 𝐹 = ( LFnl ‘ 𝑊 ) |
4 |
|
lkrss.l |
⊢ 𝐿 = ( LKer ‘ 𝑊 ) |
5 |
|
lkrss.d |
⊢ 𝐷 = ( LDual ‘ 𝑊 ) |
6 |
|
lkrss.s |
⊢ · = ( ·𝑠 ‘ 𝐷 ) |
7 |
|
lkrss.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
8 |
|
lkrss.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
9 |
|
lkrss.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐾 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
11 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
12 |
10 1 2 11 3 4 7 8 9
|
lkrscss |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) ) |
13 |
3 10 1 2 11 5 6 7 9 8
|
ldualvs |
⊢ ( 𝜑 → ( 𝑋 · 𝐺 ) = ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) |
14 |
13
|
fveq2d |
⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) = ( 𝐿 ‘ ( 𝐺 ∘f ( .r ‘ 𝑅 ) ( ( Base ‘ 𝑊 ) × { 𝑋 } ) ) ) ) |
15 |
12 14
|
sseqtrrd |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ ( 𝑋 · 𝐺 ) ) ) |