Step |
Hyp |
Ref |
Expression |
1 |
|
dih1rn.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dih1rn.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dih1rn.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dih1rn.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
6 |
5 1 2 3 4
|
dih1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 1. ‘ 𝐾 ) ) = 𝑉 ) |
7 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
8 |
7
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ OP ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
10 |
9 5
|
op1cl |
⊢ ( 𝐾 ∈ OP → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
11 |
8 10
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
12 |
9 1 2
|
dihcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( 1. ‘ 𝐾 ) ) ∈ ran 𝐼 ) |
13 |
11 12
|
mpdan |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 1. ‘ 𝐾 ) ) ∈ ran 𝐼 ) |
14 |
6 13
|
eqeltrrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑉 ∈ ran 𝐼 ) |