Step |
Hyp |
Ref |
Expression |
1 |
|
doch1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
doch1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
doch1.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
doch1.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
doch1.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
eqid |
⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
1 6 2 4
|
dih1rn |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑉 ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
8 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
9 |
8 1 6 3
|
dochvalr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ⊥ ‘ 𝑉 ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ◡ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑉 ) ) ) ) |
10 |
7 9
|
mpdan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑉 ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ◡ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑉 ) ) ) ) |
11 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
12 |
1 11 6 2 4
|
dih1cnv |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ◡ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑉 ) = ( 1. ‘ 𝐾 ) ) |
13 |
12
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( oc ‘ 𝐾 ) ‘ ( ◡ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑉 ) ) = ( ( oc ‘ 𝐾 ) ‘ ( 1. ‘ 𝐾 ) ) ) |
14 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
15 |
14
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ OP ) |
16 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
17 |
16 11 8
|
opoc1 |
⊢ ( 𝐾 ∈ OP → ( ( oc ‘ 𝐾 ) ‘ ( 1. ‘ 𝐾 ) ) = ( 0. ‘ 𝐾 ) ) |
18 |
15 17
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( oc ‘ 𝐾 ) ‘ ( 1. ‘ 𝐾 ) ) = ( 0. ‘ 𝐾 ) ) |
19 |
13 18
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( oc ‘ 𝐾 ) ‘ ( ◡ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑉 ) ) = ( 0. ‘ 𝐾 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ◡ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑉 ) ) ) = ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 0. ‘ 𝐾 ) ) ) |
21 |
16 1 6 2 5
|
dih0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 0. ‘ 𝐾 ) ) = { 0 } ) |
22 |
10 20 21
|
3eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑉 ) = { 0 } ) |