Step |
Hyp |
Ref |
Expression |
1 |
|
dihjust.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihjust.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihjust.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihjust.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
dihjust.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
dihjust.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
dihjust.i |
⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihjust.J |
⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
dihjust.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
dihjust.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
11 |
|
dihord2c.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
dihord2c.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
dihord2c.o |
⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) |
14 |
|
dihord2.p |
⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
dihord2.e |
⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
dihord2.d |
⊢ + = ( +g ‘ 𝑈 ) |
17 |
|
dihord2.g |
⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 ) |
18 |
1 2 3 4 5 6 7 8 9 10
|
dihord2a |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑄 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
19 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝐾 ∈ HL ) |
20 |
19
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝐾 ∈ Lat ) |
21 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑋 ∈ 𝐵 ) |
22 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑊 ∈ 𝐻 ) |
23 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
24 |
22 23
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑊 ∈ 𝐵 ) |
25 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
26 |
20 21 24 25
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
27 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑌 ∈ 𝐵 ) |
28 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
29 |
20 27 24 28
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
30 |
|
simp13l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑁 ∈ 𝐴 ) |
31 |
1 5
|
atbase |
⊢ ( 𝑁 ∈ 𝐴 → 𝑁 ∈ 𝐵 ) |
32 |
30 31
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑁 ∈ 𝐵 ) |
33 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑁 ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) → ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∈ 𝐵 ) |
34 |
20 32 29 33
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∈ 𝐵 ) |
35 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
36 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
|
dihord2pre |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
37 |
35 36
|
syld3an3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
38 |
1 2 3
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑁 ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
39 |
20 32 29 38
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑌 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
40 |
1 2 20 26 29 34 37 39
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
41 |
|
simp12l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑄 ∈ 𝐴 ) |
42 |
1 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
43 |
41 42
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → 𝑄 ∈ 𝐵 ) |
44 |
1 2 3
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∈ 𝐵 ) ) → ( ( 𝑄 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∧ ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ) |
45 |
20 43 26 34 44
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( ( 𝑄 ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ∧ ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ↔ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) ) |
46 |
18 40 45
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑁 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ≤ ( 𝑁 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |