Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem9.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem9.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem9.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihmeetlem9.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihmeetlem9.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihmeetlem9.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
dihmeetlem9.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihmeetlem9.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
dihmeetlem9.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
1 2 3 4 5 6 7 8 9
|
dihmeetlem10N |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ) |
11 |
10
|
ineq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
12 |
|
inass |
⊢ ( ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
13 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
14 |
13
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝐾 ∈ Lat ) |
15 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 ) |
16 |
|
simprll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑝 ∈ 𝐴 ) |
17 |
1 6
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 ) |
18 |
16 17
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑝 ∈ 𝐵 ) |
19 |
1 2 4
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) ) |
20 |
14 15 18 19
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) ) |
21 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
22 |
1 4
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( 𝑌 ∨ 𝑝 ) ∈ 𝐵 ) |
23 |
14 15 18 22
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝑌 ∨ 𝑝 ) ∈ 𝐵 ) |
24 |
1 2 3 9
|
dihord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑌 ∨ 𝑝 ) ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ↔ 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) ) ) |
25 |
21 15 23 24
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ↔ 𝑌 ≤ ( 𝑌 ∨ 𝑝 ) ) ) |
26 |
20 25
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) |
27 |
|
sseqin2 |
⊢ ( ( 𝐼 ‘ 𝑌 ) ⊆ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ↔ ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( 𝐼 ‘ 𝑌 ) ) |
28 |
26 27
|
sylib |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( 𝐼 ‘ 𝑌 ) ) |
29 |
28
|
ineq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ 𝑋 ) ∩ ( ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
30 |
12 29
|
syl5eq |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ ( 𝑌 ∨ 𝑝 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
31 |
11 30
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ 𝑝 ≤ 𝑋 ) ) → ( ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ 𝑋 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |