Step |
Hyp |
Ref |
Expression |
1 |
|
dih0rn.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dih0rn.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dih0rn.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dih0rn.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
5 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
6 |
5 1 2 3 4
|
dih0 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) = { 0 } ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
7 1 2
|
dihfn |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 Fn ( Base ‘ 𝐾 ) ) |
9 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
10 |
9
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ OP ) |
11 |
7 5
|
op0cl |
⊢ ( 𝐾 ∈ OP → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
|
fnfvelrn |
⊢ ( ( 𝐼 Fn ( Base ‘ 𝐾 ) ∧ ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) ∈ ran 𝐼 ) |
14 |
8 12 13
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝐼 ‘ ( 0. ‘ 𝐾 ) ) ∈ ran 𝐼 ) |
15 |
6 14
|
eqeltrrd |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → { 0 } ∈ ran 𝐼 ) |