Step |
Hyp |
Ref |
Expression |
1 |
|
dochsp.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochsp.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochsp.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochsp.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochsp.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
6 |
|
dochsp.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
dochsp.x |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑉 ) |
8 |
1 2 6
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
9 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
10 |
1 2 4 9 3
|
dochlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
11 |
6 7 10
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
12 |
9 5
|
lspid |
⊢ ( ( 𝑈 ∈ LMod ∧ ( ⊥ ‘ 𝑋 ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( 𝑁 ‘ ( ⊥ ‘ 𝑋 ) ) = ( ⊥ ‘ 𝑋 ) ) |
13 |
8 11 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ ( ⊥ ‘ 𝑋 ) ) = ( ⊥ ‘ 𝑋 ) ) |
14 |
1 2 3 4 5 6 7
|
dochocsp |
⊢ ( 𝜑 → ( ⊥ ‘ ( 𝑁 ‘ 𝑋 ) ) = ( ⊥ ‘ 𝑋 ) ) |
15 |
13 14
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑁 ‘ ( ⊥ ‘ 𝑋 ) ) = ( ⊥ ‘ ( 𝑁 ‘ 𝑋 ) ) ) |