Step |
Hyp |
Ref |
Expression |
1 |
|
lhpoc.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lhpoc.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
3 |
|
lhpoc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
lhpoc.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
6 |
1 2
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵 ) → ( ⊥ ‘ 𝑊 ) ∈ 𝐵 ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ) → ( ⊥ ‘ 𝑊 ) ∈ 𝐵 ) |
8 |
1 2 3 4
|
lhpoc |
⊢ ( ( 𝐾 ∈ HL ∧ ( ⊥ ‘ 𝑊 ) ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑊 ) ∈ 𝐻 ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑊 ) ) ∈ 𝐴 ) ) |
9 |
7 8
|
syldan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑊 ) ∈ 𝐻 ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑊 ) ) ∈ 𝐴 ) ) |
10 |
1 2
|
opococ |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑊 ) ) = 𝑊 ) |
11 |
5 10
|
sylan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑊 ) ) = 𝑊 ) |
12 |
11
|
eleq1d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑊 ) ) ∈ 𝐴 ↔ 𝑊 ∈ 𝐴 ) ) |
13 |
9 12
|
bitr2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ) → ( 𝑊 ∈ 𝐴 ↔ ( ⊥ ‘ 𝑊 ) ∈ 𝐻 ) ) |