Step |
Hyp |
Ref |
Expression |
1 |
|
lcdvbasecl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcdvbasecl.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcdvbasecl.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
lcdvbasecl.s |
⊢ 𝑆 = ( Scalar ‘ 𝑈 ) |
5 |
|
lcdvbasecl.r |
⊢ 𝑅 = ( Base ‘ 𝑆 ) |
6 |
|
lcdvbasecl.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
lcdvbasecl.e |
⊢ 𝐸 = ( Base ‘ 𝐶 ) |
8 |
|
lcdvbasecl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
lcdvbasecl.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐸 ) |
10 |
|
lcdvbasecl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
11 |
1 2 8
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
12 |
|
eqid |
⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 ) |
13 |
1 6 7 2 12 8 9
|
lcdvbaselfl |
⊢ ( 𝜑 → 𝐹 ∈ ( LFnl ‘ 𝑈 ) ) |
14 |
4 5 3 12
|
lflcl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ ( LFnl ‘ 𝑈 ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝑅 ) |
15 |
11 13 10 14
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ 𝑅 ) |