Step |
Hyp |
Ref |
Expression |
1 |
|
hlhil0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hlhil0.l |
⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hlhil0.u |
⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
hlhil0.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
5 |
|
hlhil0.z |
⊢ 0 = ( 0g ‘ 𝐿 ) |
6 |
|
eqidd |
⊢ ( 𝜑 → ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
8 |
1 3 4 2 7
|
hlhilbase |
⊢ ( 𝜑 → ( Base ‘ 𝐿 ) = ( Base ‘ 𝑈 ) ) |
9 |
|
eqid |
⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 ) |
10 |
1 3 4 2 9
|
hlhilplus |
⊢ ( 𝜑 → ( +g ‘ 𝐿 ) = ( +g ‘ 𝑈 ) ) |
11 |
10
|
oveqdr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐿 ) ∧ 𝑦 ∈ ( Base ‘ 𝐿 ) ) ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑈 ) 𝑦 ) ) |
12 |
6 8 11
|
grpidpropd |
⊢ ( 𝜑 → ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝑈 ) ) |
13 |
5 12
|
syl5eq |
⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑈 ) ) |