Step |
Hyp |
Ref |
Expression |
1 |
|
dih2dimb.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dih2dimb.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dih2dimb.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
dih2dimb.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
dih2dimb.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dih2dimb.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
7 |
|
dih2dimb.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dih2dimb.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
dih2dimb.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊 ) ) |
10 |
|
dih2dimb.q |
⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊 ) ) |
11 |
|
eqid |
⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
1 2 3 4 5 6 11 8 9 10
|
dib2dim |
⊢ ( 𝜑 → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ⊆ ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) ) |
13 |
8
|
simpld |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
14 |
9
|
simpld |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
15 |
10
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
17 |
16 2 3
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
13 14 15 17
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
9
|
simprd |
⊢ ( 𝜑 → 𝑃 ≤ 𝑊 ) |
20 |
10
|
simprd |
⊢ ( 𝜑 → 𝑄 ≤ 𝑊 ) |
21 |
13
|
hllatd |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
22 |
16 3
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
23 |
14 22
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
24 |
16 3
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
25 |
15 24
|
syl |
⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
26 |
8
|
simprd |
⊢ ( 𝜑 → 𝑊 ∈ 𝐻 ) |
27 |
16 4
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
28 |
26 27
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
29 |
16 1 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) ) |
30 |
21 23 25 28 29
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) ) |
31 |
19 20 30
|
mpbi2and |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) |
32 |
16 1 4 7 11
|
dihvalb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ) |
33 |
8 18 31 32
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑃 ∨ 𝑄 ) ) ) |
34 |
16 1 4 7 11
|
dihvalb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑃 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ) |
35 |
8 23 19 34
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ) |
36 |
16 1 4 7 11
|
dihvalb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) |
37 |
8 25 20 36
|
syl12anc |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑄 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) |
38 |
35 37
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) = ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑃 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑄 ) ) ) |
39 |
12 33 38
|
3sstr4d |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) ⊆ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ) |