Step |
Hyp |
Ref |
Expression |
1 |
|
dihjatc1.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihjatc1.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihjatc1.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihjatc1.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihjatc1.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihjatc1.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
dihjatc1.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihjatc1.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
9 |
|
dihjatc1.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ HL ) |
12 |
11
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ Lat ) |
13 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 ) |
14 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑌 ∈ 𝐵 ) |
15 |
1 5
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
16 |
12 13 14 15
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
17 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ∈ 𝐴 ) |
18 |
1 6
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
19 |
17 18
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ∈ 𝐵 ) |
20 |
1 4
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∈ 𝐵 ) |
21 |
12 16 19 20
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∈ 𝐵 ) |
22 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
23 |
|
simp3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ≤ 𝑋 ) |
24 |
1 2 3 4 5 6
|
dihmeetlem6 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 ) |
25 |
10 13 14 22 23 24
|
syl32anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 ) |
26 |
1 2 4 5 6
|
dihmeetlem5 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |
27 |
11 13 14 17 23 26
|
syl32anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |
28 |
27
|
breq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) ) |
29 |
25 28
|
mtbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ¬ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) |
30 |
1 2 4
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → 𝑄 ≤ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |
31 |
12 16 19 30
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑄 ≤ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |
32 |
1 2 4 5 6 3 9 7 8
|
dihvalcq2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∈ 𝐵 ∧ ¬ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) |
33 |
10 21 29 22 31 32
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) |
34 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
35 |
2 5 34 6 3
|
lhpmat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
36 |
10 22 35
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑄 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
37 |
36
|
oveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑄 ∧ 𝑊 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) ) |
38 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 ) |
39 |
1 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
40 |
38 39
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝑊 ∈ 𝐵 ) |
41 |
|
simp3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) |
42 |
1 2 4 5 6
|
atmod1i2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑄 ∧ 𝑊 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) |
43 |
11 17 16 40 41 42
|
syl131anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑄 ∧ 𝑊 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) |
44 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
45 |
11 44
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → 𝐾 ∈ OL ) |
46 |
1 4 34
|
olj01 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) ) |
47 |
45 16 46
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) ) |
48 |
37 43 47
|
3eqtr3d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑌 ) ) |
49 |
48
|
fveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) = ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) |
50 |
49
|
oveq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ∧ 𝑊 ) ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ) |
51 |
33 50
|
eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( 𝐼 ‘ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ ( 𝑋 ∧ 𝑌 ) ) ) ) |