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 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ HL ) |
12 |
11
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝐾 ∈ Lat ) |
13 |
|
simp21l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ∈ 𝐴 ) |
14 |
1 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
15 |
13 14
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ∈ 𝐵 ) |
16 |
|
simp23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑋 ∈ 𝐵 ) |
17 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐻 ) |
18 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
19 |
17 18
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑊 ∈ 𝐵 ) |
20 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
21 |
12 16 19 20
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
22 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
23 |
12 15 21 22
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
24 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
25 |
23 24
|
breqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑄 ≤ ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
26 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
27 |
|
simp22 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) |
28 |
|
simp21 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
29 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
30 |
12 16 19 29
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
31 |
21 30
|
jca |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) |
32 |
1 2 3 5 6 7 8 9 10
|
cdlemn |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) ) → ( 𝑄 ≤ ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ↔ ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
33 |
26 27 28 31 32
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝑄 ≤ ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ↔ ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
34 |
25 33
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
35 |
6 9 26
|
dvhlmod |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → 𝑈 ∈ LMod ) |
36 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
37 |
36
|
lsssssubg |
⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
38 |
35 37
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
39 |
2 5 6 9 8 36
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
40 |
26 27 39
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
41 |
38 40
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
42 |
1 2 6 9 7 36
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
43 |
26 21 30 42
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
44 |
38 43
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
45 |
10
|
lsmub2 |
⊢ ( ( ( 𝐽 ‘ 𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
46 |
41 44 45
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |
47 |
2 5 6 9 8 36
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
48 |
26 28 47
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
49 |
38 48
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
50 |
36 10
|
lsmcl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
51 |
35 40 43 50
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
52 |
38 51
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
53 |
10
|
lsmlub |
⊢ ( ( ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
54 |
49 44 52 53
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ↔ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) ) |
55 |
34 46 54
|
mpbi2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ) |