Step |
Hyp |
Ref |
Expression |
1 |
|
dochnoncon.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochnoncon.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochnoncon.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑈 ) |
4 |
|
dochnoncon.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
5 |
|
dochnoncon.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dochnel2.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
dochnel2.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
8 |
|
dochnel2.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑇 ∖ { 0 } ) ) |
9 |
8
|
eldifbd |
⊢ ( 𝜑 → ¬ 𝑋 ∈ { 0 } ) |
10 |
8
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑇 ) |
11 |
|
elin |
⊢ ( 𝑋 ∈ ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) ↔ ( 𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) ) |
12 |
1 2 3 4 5
|
dochnoncon |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑇 ∈ 𝑆 ) → ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) = { 0 } ) |
13 |
6 7 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) = { 0 } ) |
14 |
13
|
eleq2d |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑇 ∩ ( ⊥ ‘ 𝑇 ) ) ↔ 𝑋 ∈ { 0 } ) ) |
15 |
11 14
|
bitr3id |
⊢ ( 𝜑 → ( ( 𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) ↔ 𝑋 ∈ { 0 } ) ) |
16 |
15
|
biimpd |
⊢ ( 𝜑 → ( ( 𝑋 ∈ 𝑇 ∧ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) → 𝑋 ∈ { 0 } ) ) |
17 |
10 16
|
mpand |
⊢ ( 𝜑 → ( 𝑋 ∈ ( ⊥ ‘ 𝑇 ) → 𝑋 ∈ { 0 } ) ) |
18 |
9 17
|
mtod |
⊢ ( 𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘ 𝑇 ) ) |