Step |
Hyp |
Ref |
Expression |
1 |
|
dihoml4.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dihoml4.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dihoml4.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑈 ) |
4 |
|
dihoml4.o |
⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dihoml4.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
dihoml4.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑆 ) |
7 |
|
dihoml4.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑆 ) |
8 |
|
dihoml4.c |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 ) |
9 |
|
dihoml4.l |
⊢ ( 𝜑 → 𝑋 ⊆ 𝑌 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
11 |
10 3
|
lssss |
⊢ ( 𝑋 ∈ 𝑆 → 𝑋 ⊆ ( Base ‘ 𝑈 ) ) |
12 |
6 11
|
syl |
⊢ ( 𝜑 → 𝑋 ⊆ ( Base ‘ 𝑈 ) ) |
13 |
|
eqid |
⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
14 |
1 13 2 10 4
|
dochcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ ( Base ‘ 𝑈 ) ) → ( ⊥ ‘ 𝑋 ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
15 |
5 12 14
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ 𝑋 ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
16 |
1 13 4
|
dochoc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑋 ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) ) |
17 |
5 15 16
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) = ( ⊥ ‘ 𝑋 ) ) |
18 |
17
|
ineq1d |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) ∩ 𝑌 ) = ( ( ⊥ ‘ 𝑋 ) ∩ 𝑌 ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝜑 → ( ⊥ ‘ ( ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) ∩ 𝑌 ) ) = ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑌 ) ) ) |
20 |
19
|
ineq1d |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) ∩ 𝑌 ) ) ∩ 𝑌 ) = ( ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑌 ) ) ∩ 𝑌 ) ) |
21 |
1 2 10 4
|
dochssv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ ( Base ‘ 𝑈 ) ) → ( ⊥ ‘ 𝑋 ) ⊆ ( Base ‘ 𝑈 ) ) |
22 |
5 12 21
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ 𝑋 ) ⊆ ( Base ‘ 𝑈 ) ) |
23 |
1 13 2 10 4
|
dochcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑋 ) ⊆ ( Base ‘ 𝑈 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
24 |
5 22 23
|
syl2anc |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
25 |
10 3
|
lssss |
⊢ ( 𝑌 ∈ 𝑆 → 𝑌 ⊆ ( Base ‘ 𝑈 ) ) |
26 |
7 25
|
syl |
⊢ ( 𝜑 → 𝑌 ⊆ ( Base ‘ 𝑈 ) ) |
27 |
1 13 2 10 4 5 26
|
dochoccl |
⊢ ( 𝜑 → ( 𝑌 ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 ) ) |
28 |
8 27
|
mpbird |
⊢ ( 𝜑 → 𝑌 ∈ ran ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ) |
29 |
1 2 10 4
|
dochss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ⊆ ( Base ‘ 𝑈 ) ∧ 𝑋 ⊆ 𝑌 ) → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) |
30 |
5 26 9 29
|
syl3anc |
⊢ ( 𝜑 → ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) |
31 |
1 2 10 4
|
dochss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ⊥ ‘ 𝑋 ) ⊆ ( Base ‘ 𝑈 ) ∧ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) |
32 |
5 22 30 31
|
syl3anc |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) ) |
33 |
32 8
|
sseqtrd |
⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ⊆ 𝑌 ) |
34 |
1 13 4 5 24 28 33
|
dihoml4c |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ( ⊥ ‘ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) ∩ 𝑌 ) ) ∩ 𝑌 ) = ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) |
35 |
20 34
|
eqtr3d |
⊢ ( 𝜑 → ( ( ⊥ ‘ ( ( ⊥ ‘ 𝑋 ) ∩ 𝑌 ) ) ∩ 𝑌 ) = ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) ) |