Step |
Hyp |
Ref |
Expression |
1 |
|
lcdvaddval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
lcdvaddval.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
lcdvaddval.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
lcdvaddval.r |
⊢ 𝑅 = ( Scalar ‘ 𝑈 ) |
5 |
|
lcdvaddval.a |
⊢ + = ( +g ‘ 𝑅 ) |
6 |
|
lcdvaddval.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
lcdvaddval.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
8 |
|
lcdvaddval.p |
⊢ ✚ = ( +g ‘ 𝐶 ) |
9 |
|
lcdvaddval.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
lcdvaddval.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
11 |
|
lcdvaddval.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐷 ) |
12 |
|
lcdvaddval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
13 |
|
eqid |
⊢ ( LDual ‘ 𝑈 ) = ( LDual ‘ 𝑈 ) |
14 |
|
eqid |
⊢ ( +g ‘ ( LDual ‘ 𝑈 ) ) = ( +g ‘ ( LDual ‘ 𝑈 ) ) |
15 |
1 2 13 14 6 8 9
|
lcdvadd |
⊢ ( 𝜑 → ✚ = ( +g ‘ ( LDual ‘ 𝑈 ) ) ) |
16 |
15
|
oveqd |
⊢ ( 𝜑 → ( 𝐹 ✚ 𝐺 ) = ( 𝐹 ( +g ‘ ( LDual ‘ 𝑈 ) ) 𝐺 ) ) |
17 |
16
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ✚ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ( +g ‘ ( LDual ‘ 𝑈 ) ) 𝐺 ) ‘ 𝑋 ) ) |
18 |
|
eqid |
⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 ) |
19 |
1 2 9
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
20 |
1 6 7 2 18 9 10
|
lcdvbaselfl |
⊢ ( 𝜑 → 𝐹 ∈ ( LFnl ‘ 𝑈 ) ) |
21 |
1 6 7 2 18 9 11
|
lcdvbaselfl |
⊢ ( 𝜑 → 𝐺 ∈ ( LFnl ‘ 𝑈 ) ) |
22 |
3 4 5 18 13 14 19 20 21 12
|
ldualvaddval |
⊢ ( 𝜑 → ( ( 𝐹 ( +g ‘ ( LDual ‘ 𝑈 ) ) 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑋 ) + ( 𝐺 ‘ 𝑋 ) ) ) |
23 |
17 22
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐹 ✚ 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑋 ) + ( 𝐺 ‘ 𝑋 ) ) ) |