Step |
Hyp |
Ref |
Expression |
1 |
|
dochnel.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochnel.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochnel.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochnel.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochnel.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
dochnel.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
dochnel.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
8 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
9 |
1 3 6
|
dvhlmod |
⊢ ( 𝜑 → 𝑈 ∈ LMod ) |
10 |
7
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
11 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
12 |
4 8 11
|
lspsncl |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
13 |
9 10 12
|
syl2anc |
⊢ ( 𝜑 → ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) |
14 |
4 11
|
lspsnid |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) |
15 |
9 10 14
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) |
16 |
|
eldifsni |
⊢ ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) → 𝑋 ≠ 0 ) |
17 |
7 16
|
syl |
⊢ ( 𝜑 → 𝑋 ≠ 0 ) |
18 |
|
eldifsn |
⊢ ( 𝑋 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∖ { 0 } ) ↔ ( 𝑋 ∈ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∧ 𝑋 ≠ 0 ) ) |
19 |
15 17 18
|
sylanbrc |
⊢ ( 𝜑 → 𝑋 ∈ ( ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ∖ { 0 } ) ) |
20 |
1 3 8 5 2 6 13 19
|
dochnel2 |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) ) |
21 |
10
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 ) |
22 |
1 3 2 4 11 6 21
|
dochocsp |
⊢ ( 𝜑 → ( ⊥ ‘ ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = ( ⊥ ‘ { 𝑋 } ) ) |
23 |
20 22
|
neleqtrd |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘ { 𝑋 } ) ) |