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 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) |
13 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
14 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝐾 ∈ HL ) |
15 |
14
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝐾 ∈ Lat ) |
16 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
17 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 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 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑊 ∈ 𝐵 ) |
20 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
21 |
15 16 19 20
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) |
22 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
23 |
15 16 19 22
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) |
24 |
21 23
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) |
25 |
|
simp12l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑄 ∈ 𝐴 ) |
26 |
1 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
27 |
25 26
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑄 ∈ 𝐵 ) |
28 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
29 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
30 |
15 28 19 29
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
31 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝐵 ) |
32 |
15 27 30 31
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ∈ 𝐵 ) |
33 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
34 |
15 27 30 33
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
35 |
|
simp31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) |
36 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑋 ≤ 𝑌 ) |
37 |
35 36
|
eqbrtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) ≤ 𝑌 ) |
38 |
1 2 15 27 32 16 34 37
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑄 ≤ 𝑌 ) |
39 |
|
simp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) |
40 |
38 39
|
breqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑄 ≤ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) ) |
41 |
1 2 3 5 6 9 10 7 8
|
cdlemn5 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) ∧ 𝑄 ≤ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) ) → ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
42 |
11 12 13 24 40 41
|
syl131anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
43 |
1 2 4
|
latmlem1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
44 |
15 28 16 19 43
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
45 |
36 44
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) |
46 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
47 |
15 28 19 46
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
48 |
1 2 6 7
|
dibord |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ↔ ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
49 |
11 30 47 21 23 48
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ↔ ( 𝑋 ∧ 𝑊 ) ≤ ( 𝑌 ∧ 𝑊 ) ) ) |
50 |
45 49
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) |
51 |
6 9 11
|
dvhlmod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → 𝑈 ∈ LMod ) |
52 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
53 |
52
|
lsssssubg |
⊢ ( 𝑈 ∈ LMod → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
54 |
51 53
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( LSubSp ‘ 𝑈 ) ⊆ ( SubGrp ‘ 𝑈 ) ) |
55 |
2 5 6 9 8 52
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
56 |
11 12 55
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
57 |
54 56
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐽 ‘ 𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
58 |
1 2 6 9 7 52
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
59 |
11 21 23 58
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
60 |
54 59
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
61 |
10
|
lsmub2 |
⊢ ( ( ( 𝐽 ‘ 𝑅 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
62 |
57 60 61
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
63 |
50 62
|
sstrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |
64 |
2 5 6 9 8 52
|
diclss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
65 |
11 13 64
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( LSubSp ‘ 𝑈 ) ) |
66 |
54 65
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ) |
67 |
1 2 6 9 7 52
|
diblss |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
68 |
11 30 47 67
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
69 |
54 68
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
70 |
52 10
|
lsmcl |
⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑅 ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
71 |
51 56 59 70
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∈ ( LSubSp ‘ 𝑈 ) ) |
72 |
54 71
|
sseldd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) ) |
73 |
10
|
lsmlub |
⊢ ( ( ( 𝐽 ‘ 𝑄 ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ∈ ( SubGrp ‘ 𝑈 ) ∧ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∈ ( SubGrp ‘ 𝑈 ) ) → ( ( ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ↔ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
74 |
66 69 72 73
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( ( 𝐽 ‘ 𝑄 ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ∧ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ↔ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) ) |
75 |
42 63 74
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ∧ 𝑋 ≤ 𝑌 ) ) → ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) ⊆ ( ( 𝐽 ‘ 𝑅 ) ⊕ ( 𝐼 ‘ ( 𝑌 ∧ 𝑊 ) ) ) ) |