Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem14.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem14.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem14.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihmeetlem14.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihmeetlem14.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihmeetlem14.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
dihmeetlem14.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihmeetlem14.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
dihmeetlem14.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
dihmeetlem15.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
11 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
simpr1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ) |
13 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) |
14 |
|
simpl3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ¬ 𝑝 ≤ 𝑊 ) |
15 |
|
simp3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑟 = 𝑝 ) |
16 |
|
simp22 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑟 ≤ 𝑌 ) |
17 |
15 16
|
eqbrtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ≤ 𝑌 ) |
18 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝐾 ∈ HL ) |
19 |
18
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝐾 ∈ Lat ) |
20 |
|
simp13l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ∈ 𝐴 ) |
21 |
1 6
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 ) |
22 |
20 21
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ∈ 𝐵 ) |
23 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑌 ∈ 𝐵 ) |
24 |
1 2 5
|
latleeqm2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑌 ∧ 𝑝 ) = 𝑝 ) ) |
25 |
19 22 23 24
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑌 ∧ 𝑝 ) = 𝑝 ) ) |
26 |
17 25
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑌 ∧ 𝑝 ) = 𝑝 ) |
27 |
|
simp23 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) |
28 |
26 27
|
eqbrtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ∧ 𝑟 = 𝑝 ) → 𝑝 ≤ 𝑊 ) |
29 |
28
|
3expia |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( 𝑟 = 𝑝 → 𝑝 ≤ 𝑊 ) ) |
30 |
29
|
necon3bd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( ¬ 𝑝 ≤ 𝑊 → 𝑟 ≠ 𝑝 ) ) |
31 |
14 30
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → 𝑟 ≠ 𝑝 ) |
32 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
33 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
34 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
35 |
|
eqid |
⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) |
36 |
|
eqid |
⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑟 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑟 ) |
37 |
|
eqid |
⊢ ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑝 ) = ( ℩ ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( ℎ ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑝 ) |
38 |
1 2 4 6 3 32 33 34 35 9 7 10 36 37
|
dihmeetlem13N |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ 𝑟 ≠ 𝑝 ) → ( ( 𝐼 ‘ 𝑟 ) ∩ ( 𝐼 ‘ 𝑝 ) ) = { 0 } ) |
39 |
11 12 13 31 38
|
syl121anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝑟 ≤ 𝑌 ∧ ( 𝑌 ∧ 𝑝 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑟 ) ∩ ( 𝐼 ‘ 𝑝 ) ) = { 0 } ) |