Step |
Hyp |
Ref |
Expression |
1 |
|
dochsscl.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dochsscl.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dochsscl.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
dochsscl.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dochsscl.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dochsscl.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
7 |
|
dochsscl.x |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑉 ) |
8 |
|
dochsscl.y |
⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 ) |
9 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → 𝑋 ⊆ 𝑉 ) |
11 |
1 2 3 5
|
dochssv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝑉 ) |
12 |
9 10 11
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑋 ) ⊆ 𝑉 ) |
13 |
1 2 4 3
|
dihrnss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → 𝑌 ⊆ 𝑉 ) |
14 |
6 8 13
|
syl2anc |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑉 ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → 𝑌 ⊆ 𝑉 ) |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → 𝑋 ⊆ 𝑌 ) |
17 |
1 2 3 5
|
dochss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) |
18 |
9 15 16 17
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) |
19 |
1 2 3 5
|
dochss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑋 ) ⊆ 𝑉 ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) |
20 |
9 12 18 19
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) |
21 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → 𝑌 ∈ ran 𝐼 ) |
22 |
1 4 5
|
dochoc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ ran 𝐼 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 ) |
23 |
9 21 22
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 ) |
24 |
20 23
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ 𝑌 ) |
25 |
1 2 3 5
|
dochocss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑉 ) → 𝑋 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) |
26 |
6 7 25
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) |
27 |
|
sstr |
⊢ ( ( 𝑋 ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ 𝑌 ) → 𝑋 ⊆ 𝑌 ) |
28 |
26 27
|
sylan |
⊢ ( ( 𝜑 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ 𝑌 ) → 𝑋 ⊆ 𝑌 ) |
29 |
24 28
|
impbida |
⊢ ( 𝜑 → ( 𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ 𝑌 ) ) |