Step |
Hyp |
Ref |
Expression |
1 |
|
hlhillvec.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hlhillvec.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hlhillvec.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
4 |
|
eqid |
⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
1 4 3
|
dvhlvec |
⊢ ( 𝜑 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec ) |
6 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
7 |
|
eqid |
⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
8 |
1 2 3 4 7
|
hlhilbase |
⊢ ( 𝜑 → ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ 𝑈 ) ) |
9 |
|
eqid |
⊢ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
10 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
11 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) |
12 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
13 |
1 4 9 2 10 3 12
|
hlhilsbase2 |
⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
14 |
|
eqid |
⊢ ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
15 |
1 2 3 4 14
|
hlhilplus |
⊢ ( 𝜑 → ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ 𝑈 ) ) |
16 |
15
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( 𝑥 ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑈 ) 𝑦 ) ) |
17 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( +g ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
18 |
1 4 9 2 10 3 17
|
hlhilsplus2 |
⊢ ( 𝜑 → ( +g ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) ) ) |
19 |
18
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ 𝑦 ∈ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) ) → ( 𝑥 ( +g ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) 𝑦 ) = ( 𝑥 ( +g ‘ ( Scalar ‘ 𝑈 ) ) 𝑦 ) ) |
20 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( .r ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
21 |
1 4 9 2 10 3 20
|
hlhilsmul2 |
⊢ ( 𝜑 → ( .r ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( .r ‘ ( Scalar ‘ 𝑈 ) ) ) |
22 |
21
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ 𝑦 ∈ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) ) → ( 𝑥 ( .r ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) 𝑦 ) = ( 𝑥 ( .r ‘ ( Scalar ‘ 𝑈 ) ) 𝑦 ) ) |
23 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
24 |
1 4 23 2 3
|
hlhilvsca |
⊢ ( 𝜑 → ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ 𝑈 ) ) |
25 |
24
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∧ 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( 𝑥 ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝑈 ) 𝑦 ) ) |
26 |
6 8 9 10 11 13 16 19 22 25
|
lvecprop2d |
⊢ ( 𝜑 → ( ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec ↔ 𝑈 ∈ LVec ) ) |
27 |
5 26
|
mpbid |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |