Step |
Hyp |
Ref |
Expression |
1 |
|
dochoc1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochoc1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochoc1.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochoc1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochoc1.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
eqid |
⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 ) |
7 |
1 2 3 4 6
|
doch1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑉 ) = { ( 0g ‘ 𝑈 ) } ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = ( ⊥ ‘ { ( 0g ‘ 𝑈 ) } ) ) |
9 |
1 2 3 4 6
|
doch0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ { ( 0g ‘ 𝑈 ) } ) = 𝑉 ) |
10 |
8 9
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = 𝑉 ) |
11 |
5 10
|
syl |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑉 ) ) = 𝑉 ) |