Step |
Hyp |
Ref |
Expression |
1 |
|
dihord3.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihord3.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihord3.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihord3.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) |
7 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
10 |
1 2 7 8 9 3
|
lhpmcvr2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) → ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) ) |
11 |
5 6 10
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) ) |
12 |
|
simp1r |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑌 ) |
13 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ≤ 𝑊 ) |
14 |
13
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ 𝑊 ) |
15 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝐾 ∈ HL ) |
16 |
15
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ HL ) |
17 |
16
|
hllatd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ Lat ) |
18 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ∈ 𝐵 ) |
19 |
18
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ∈ 𝐵 ) |
20 |
|
simpl3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑌 ∈ 𝐵 ) |
21 |
20
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑌 ∈ 𝐵 ) |
22 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → 𝑊 ∈ 𝐻 ) |
23 |
22
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑊 ∈ 𝐻 ) |
24 |
1 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
25 |
23 24
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑊 ∈ 𝐵 ) |
26 |
1 2 8
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊 ) ↔ 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
27 |
17 19 21 25 26
|
syl13anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊 ) ↔ 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
28 |
12 14 27
|
mpbi2and |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) |
29 |
|
simp1l1 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
30 |
|
simp1l2 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) |
31 |
1 8
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ) |
32 |
17 21 25 31
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ) |
33 |
1 2 8
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 ) |
34 |
17 21 25 33
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 ) |
35 |
|
eqid |
⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) |
36 |
1 2 3 35
|
dibord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 ) ) → ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ↔ 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
37 |
29 30 32 34 36
|
syl112anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ↔ 𝑋 ≤ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
38 |
28 37
|
mpbird |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
39 |
|
eqid |
⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
40 |
3 39 29
|
dvhlmod |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ) |
41 |
|
eqid |
⊢ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
42 |
41
|
lsssssubg |
⊢ ( ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod → ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⊆ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
43 |
40 42
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⊆ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
44 |
|
simp2 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ) |
45 |
|
eqid |
⊢ ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) |
46 |
2 9 3 39 45 41
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
47 |
29 44 46
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
48 |
43 47
|
sseldd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
49 |
1 2 3 39 35 41
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
50 |
29 32 34 49
|
syl12anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
51 |
43 50
|
sseldd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
52 |
|
eqid |
⊢ ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
53 |
52
|
lsmub2 |
⊢ ( ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ( SubGrp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
54 |
48 51 53
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
55 |
38 54
|
sstrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ⊆ ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
56 |
1 2 3 4 35
|
dihvalb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) |
57 |
29 30 56
|
syl2anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) |
58 |
|
simp1l3 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) |
59 |
|
simp3 |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) |
60 |
1 2 7 8 9 3 4 35 45 39 52
|
dihvalcq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ∧ ( ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
61 |
29 58 44 59 60
|
syl112anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑌 ) = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑟 ) ( LSSum ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) |
62 |
55 57 61
|
3sstr4d |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) ∧ ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
63 |
62
|
3exp |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑟 ≤ 𝑊 ) → ( ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) ) |
64 |
63
|
expd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) → ( ¬ 𝑟 ≤ 𝑊 → ( ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) ) ) |
65 |
64
|
imp4a |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑟 ∈ ( Atoms ‘ 𝐾 ) → ( ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) ) |
66 |
65
|
rexlimdv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( ∃ 𝑟 ∈ ( Atoms ‘ 𝐾 ) ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑟 ( join ‘ 𝐾 ) ( 𝑌 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) ) |
67 |
11 66
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊 ) ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝐼 ‘ 𝑋 ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |