Step |
Hyp |
Ref |
Expression |
1 |
|
dihord5apre.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihord5apre.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihord5apre.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihord5apre.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihord5apre.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihord5apre.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
dihord5apre.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihord5apre.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
dihord5apre.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) |
12 |
1 2 4 5 6 3
|
lhpmcvr2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) |
13 |
10 11 12
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) |
14 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝐾 ∈ HL ) |
15 |
14
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝐾 ∈ Lat ) |
16 |
|
simp12l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
17 |
|
simp3ll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑟 ∈ 𝐴 ) |
18 |
1 6
|
atbase |
⊢ ( 𝑟 ∈ 𝐴 → 𝑟 ∈ 𝐵 ) |
19 |
17 18
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑟 ∈ 𝐵 ) |
20 |
1 4
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ) |
21 |
15 19 16 20
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ) |
22 |
|
simp13l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
23 |
1 2 4
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ ( 𝑟 ∨ 𝑋 ) ) |
24 |
15 19 16 23
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑋 ≤ ( 𝑟 ∨ 𝑋 ) ) |
25 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
26 |
|
simp3lr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ¬ 𝑟 ≤ 𝑊 ) |
27 |
1 2 4
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑟 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) ) |
28 |
15 19 16 27
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) ) |
29 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑊 ∈ 𝐻 ) |
30 |
1 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
31 |
29 30
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑊 ∈ 𝐵 ) |
32 |
1 2
|
lattr |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∈ 𝐵 ∧ ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) ∧ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) → 𝑟 ≤ 𝑊 ) ) |
33 |
15 19 21 31 32
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ≤ ( 𝑟 ∨ 𝑋 ) ∧ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) → 𝑟 ≤ 𝑊 ) ) |
34 |
28 33
|
mpand |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 → 𝑟 ≤ 𝑊 ) ) |
35 |
26 34
|
mtod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ¬ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) |
36 |
|
simp3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ) |
37 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) |
38 |
1 2 4 5 6 3
|
lhple |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) = 𝑋 ) |
39 |
25 36 37 38
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) = 𝑋 ) |
40 |
39
|
oveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( 𝑟 ∨ 𝑋 ) ) |
41 |
|
eqid |
⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
42 |
|
eqid |
⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
43 |
1 2 4 5 6 3 9 41 42 7 8
|
dihvalcq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( 𝑟 ∨ 𝑋 ) ) ) → ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ) |
44 |
25 21 35 36 40 43
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ) |
45 |
3 7 25
|
dvhlmod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑈 ∈ LMod ) |
46 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
47 |
46
|
lsssssubg |
⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
48 |
45 47
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
49 |
2 6 3 7 42 46
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
50 |
25 36 49
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
51 |
48 50
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
52 |
1 5
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
53 |
15 22 31 52
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
54 |
1 2 5
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
55 |
15 22 31 54
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
56 |
1 2 3 7 41 46
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
57 |
25 53 55 56
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
58 |
48 57
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
59 |
8
|
lsmub1 |
⊢ ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
60 |
51 58 59
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
61 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) |
62 |
|
simp3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) |
63 |
1 2 4 5 6 3 9 41 42 7 8
|
dihvalcq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
64 |
25 61 36 62 63
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
65 |
60 64
|
sseqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
66 |
39
|
fveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) |
67 |
1 2 3 9 41
|
dihvalb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) |
68 |
25 37 67
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) |
69 |
66 68
|
eqtr4d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) = ( 𝐼 ‘ 𝑋 ) ) |
70 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
71 |
69 70
|
eqsstrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
72 |
1 5
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ∈ 𝐵 ) |
73 |
15 21 31 72
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ∈ 𝐵 ) |
74 |
1 2 5
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ≤ 𝑊 ) |
75 |
15 21 31 74
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ≤ 𝑊 ) |
76 |
1 2 3 7 41 46
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ∈ 𝐵 ∧ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
77 |
25 73 75 76
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
78 |
48 77
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
79 |
1 3 9 7 46
|
dihlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
80 |
25 22 79
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
81 |
48 80
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
82 |
8
|
lsmlub |
⊢ ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ↔ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) |
83 |
51 78 81 82
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ↔ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) |
84 |
65 71 83
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ⊕ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( 𝑟 ∨ 𝑋 ) ∧ 𝑊 ) ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
85 |
44 84
|
eqsstrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
86 |
1 2 3 9
|
dihord4 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∨ 𝑋 ) ∈ 𝐵 ∧ ¬ ( 𝑟 ∨ 𝑋 ) ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑟 ∨ 𝑋 ) ≤ 𝑌 ) ) |
87 |
25 21 35 61 86
|
syl121anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑟 ∨ 𝑋 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑟 ∨ 𝑋 ) ≤ 𝑌 ) ) |
88 |
85 87
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑟 ∨ 𝑋 ) ≤ 𝑌 ) |
89 |
1 2 15 16 21 22 24 88
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑋 ≤ 𝑌 ) |
90 |
89
|
3expia |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) ) |
91 |
90
|
exp4c |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝑟 ∈ 𝐴 → ( ¬ 𝑟 ≤ 𝑊 → ( ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 → 𝑋 ≤ 𝑌 ) ) ) ) |
92 |
91
|
imp4a |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( 𝑟 ∈ 𝐴 → ( ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) ) ) |
93 |
92
|
rexlimdv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) ) |
94 |
13 93
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → 𝑋 ≤ 𝑌 ) |