Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem9.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem9.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem9.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihmeetlem9.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihmeetlem9.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihmeetlem9.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
dihmeetlem9.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihmeetlem9.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
dihmeetlem9.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
3 7 10
|
dvhlmod |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑈 ∈ LMod ) |
12 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
13 |
12
|
lsssssubg |
⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
14 |
11 13
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
15 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ HL ) |
16 |
15
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ Lat ) |
17 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 ) |
18 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 ) |
19 |
1 5
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
20 |
16 17 18 19
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
21 |
1 3 9 7 12
|
dihlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
22 |
10 20 21
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
23 |
14 22
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
24 |
1 6
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 ) |
25 |
24
|
3ad2ant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ∈ 𝐵 ) |
26 |
1 3 9 7 12
|
dihlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑝 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑝 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
27 |
10 25 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑝 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
28 |
14 27
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
29 |
1 3 9 7 12
|
dihlss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑌 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
30 |
10 18 29
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
31 |
14 30
|
sseldd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
32 |
1 2 5
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) |
33 |
16 17 18 32
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) |
34 |
1 2 3 9
|
dihord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ) |
35 |
10 20 18 34
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ) |
36 |
33 35
|
mpbird |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) |
37 |
8
|
lsmmod |
⊢ ( ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑌 ) ∈ ( SubGrp ‘ 𝑈 ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊆ ( 𝐼 ‘ 𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
38 |
23 28 31 36 37
|
syl31anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) = ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
39 |
|
lmodabl |
⊢ ( 𝑈 ∈ LMod → 𝑈 ∈ Abel ) |
40 |
11 39
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑈 ∈ Abel ) |
41 |
8
|
lsmcom |
⊢ ( ( 𝑈 ∈ Abel ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ 𝑝 ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ) |
42 |
40 23 28 41
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) = ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ) |
43 |
42
|
ineq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( 𝐼 ‘ 𝑝 ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) |
44 |
38 43
|
eqtr2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ( 𝐼 ‘ 𝑝 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ∩ ( 𝐼 ‘ 𝑌 ) ) = ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ⊕ ( ( 𝐼 ‘ 𝑝 ) ∩ ( 𝐼 ‘ 𝑌 ) ) ) ) |