Step |
Hyp |
Ref |
Expression |
1 |
|
dihsslss.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dihsslss.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dihsslss.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dihsslss.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑈 ) |
5 |
1 3
|
dihcnvid2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) = 𝑥 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
6 1 3
|
dihcnvcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ) |
8 |
6 1 3 2 4
|
dihlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ◡ 𝐼 ‘ 𝑥 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) ∈ 𝑆 ) |
9 |
7 8
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → ( 𝐼 ‘ ( ◡ 𝐼 ‘ 𝑥 ) ) ∈ 𝑆 ) |
10 |
5 9
|
eqeltrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ ran 𝐼 ) → 𝑥 ∈ 𝑆 ) |
11 |
10
|
ex |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑥 ∈ ran 𝐼 → 𝑥 ∈ 𝑆 ) ) |
12 |
11
|
ssrdv |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ran 𝐼 ⊆ 𝑆 ) |