Step |
Hyp |
Ref |
Expression |
1 |
|
dochsnkr2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochsnkr2.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochsnkr2.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dochsnkr2.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
dochsnkr2.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
dochsnkr2.a |
⊢ + = ( +g ‘ 𝑈 ) |
7 |
|
dochsnkr2.t |
⊢ · = ( ·𝑠 ‘ 𝑈 ) |
8 |
|
dochsnkr2.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
9 |
|
dochsnkr2.d |
⊢ 𝐷 = ( Scalar ‘ 𝑈 ) |
10 |
|
dochsnkr2.r |
⊢ 𝑅 = ( Base ‘ 𝐷 ) |
11 |
|
dochsnkr2.g |
⊢ 𝐺 = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑋 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑋 ) ) ) ) |
12 |
|
dochsnkr2.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
|
dochsnkr2.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
14 |
|
eqid |
⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 ) |
15 |
|
eqid |
⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 ) |
16 |
|
eqid |
⊢ ( LSHyp ‘ 𝑈 ) = ( LSHyp ‘ 𝑈 ) |
17 |
1 3 12
|
dvhlvec |
⊢ ( 𝜑 → 𝑈 ∈ LVec ) |
18 |
1 2 3 4 5 16 12 13
|
dochsnshp |
⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 } ) ∈ ( LSHyp ‘ 𝑈 ) ) |
19 |
13
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
20 |
1 2 3 4 5 14 15 12 13
|
dochexmidat |
⊢ ( 𝜑 → ( ( ⊥ ‘ { 𝑋 } ) ( LSSum ‘ 𝑈 ) ( ( LSpan ‘ 𝑈 ) ‘ { 𝑋 } ) ) = 𝑉 ) |
21 |
4 6 14 15 16 17 18 19 20 9 10 7 11 8
|
lshpkr |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( ⊥ ‘ { 𝑋 } ) ) |