Step |
Hyp |
Ref |
Expression |
1 |
|
latdisd.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
latdisd.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
latdisd.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
1 3
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∧ 𝑦 ) ∈ 𝐵 ) |
5 |
4
|
3adant3r3 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∧ 𝑦 ) ∈ 𝐵 ) |
6 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
7 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑧 ∈ 𝐵 ) |
8 |
|
oveq1 |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑦 ) → ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑣 ∧ 𝑤 ) ) ) |
9 |
|
oveq1 |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑦 ) → ( 𝑢 ∨ 𝑣 ) = ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑣 ) ) |
10 |
|
oveq1 |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑦 ) → ( 𝑢 ∨ 𝑤 ) = ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) |
11 |
9 10
|
oveq12d |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑦 ) → ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑣 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) ) |
12 |
8 11
|
eqeq12d |
⊢ ( 𝑢 = ( 𝑥 ∧ 𝑦 ) → ( ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ↔ ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑣 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) ) ) |
13 |
|
oveq1 |
⊢ ( 𝑣 = 𝑥 → ( 𝑣 ∧ 𝑤 ) = ( 𝑥 ∧ 𝑤 ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝑣 = 𝑥 → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑣 = 𝑥 → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑣 ) = ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ) |
16 |
15
|
oveq1d |
⊢ ( 𝑣 = 𝑥 → ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑣 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) ) |
17 |
14 16
|
eqeq12d |
⊢ ( 𝑣 = 𝑥 → ( ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑣 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) ↔ ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) ) ) |
18 |
|
oveq2 |
⊢ ( 𝑤 = 𝑧 → ( 𝑥 ∧ 𝑤 ) = ( 𝑥 ∧ 𝑧 ) ) |
19 |
18
|
oveq2d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) |
20 |
|
oveq2 |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) = ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑤 = 𝑧 → ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) ) |
22 |
19 21
|
eqeq12d |
⊢ ( 𝑤 = 𝑧 → ( ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑤 ) ) ↔ ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) ) ) |
23 |
12 17 22
|
rspc3v |
⊢ ( ( ( 𝑥 ∧ 𝑦 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) ) ) |
24 |
5 6 7 23
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) ) ) |
25 |
24
|
imp |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) = ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) ) |
26 |
|
simpl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝐾 ∈ Lat ) |
27 |
1 2
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∧ 𝑦 ) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) = ( 𝑥 ∨ ( 𝑥 ∧ 𝑦 ) ) ) |
28 |
26 5 6 27
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) = ( 𝑥 ∨ ( 𝑥 ∧ 𝑦 ) ) ) |
29 |
1 2 3
|
latabs1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∨ ( 𝑥 ∧ 𝑦 ) ) = 𝑥 ) |
30 |
29
|
3adant3r3 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∨ ( 𝑥 ∧ 𝑦 ) ) = 𝑥 ) |
31 |
28 30
|
eqtrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) = 𝑥 ) |
32 |
1 2
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∧ 𝑦 ) ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) = ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) |
33 |
26 5 7 32
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) = ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) |
34 |
31 33
|
oveq12d |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) = ( 𝑥 ∧ ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) ) |
35 |
34
|
adantr |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑥 ) ∧ ( ( 𝑥 ∧ 𝑦 ) ∨ 𝑧 ) ) = ( 𝑥 ∧ ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) ) |
36 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
37 |
|
oveq1 |
⊢ ( 𝑢 = 𝑧 → ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( 𝑧 ∨ ( 𝑣 ∧ 𝑤 ) ) ) |
38 |
|
oveq1 |
⊢ ( 𝑢 = 𝑧 → ( 𝑢 ∨ 𝑣 ) = ( 𝑧 ∨ 𝑣 ) ) |
39 |
|
oveq1 |
⊢ ( 𝑢 = 𝑧 → ( 𝑢 ∨ 𝑤 ) = ( 𝑧 ∨ 𝑤 ) ) |
40 |
38 39
|
oveq12d |
⊢ ( 𝑢 = 𝑧 → ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑣 ) ∧ ( 𝑧 ∨ 𝑤 ) ) ) |
41 |
37 40
|
eqeq12d |
⊢ ( 𝑢 = 𝑧 → ( ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ↔ ( 𝑧 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑣 ) ∧ ( 𝑧 ∨ 𝑤 ) ) ) ) |
42 |
13
|
oveq2d |
⊢ ( 𝑣 = 𝑥 → ( 𝑧 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( 𝑧 ∨ ( 𝑥 ∧ 𝑤 ) ) ) |
43 |
|
oveq2 |
⊢ ( 𝑣 = 𝑥 → ( 𝑧 ∨ 𝑣 ) = ( 𝑧 ∨ 𝑥 ) ) |
44 |
43
|
oveq1d |
⊢ ( 𝑣 = 𝑥 → ( ( 𝑧 ∨ 𝑣 ) ∧ ( 𝑧 ∨ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑤 ) ) ) |
45 |
42 44
|
eqeq12d |
⊢ ( 𝑣 = 𝑥 → ( ( 𝑧 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑣 ) ∧ ( 𝑧 ∨ 𝑤 ) ) ↔ ( 𝑧 ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑤 ) ) ) ) |
46 |
|
oveq2 |
⊢ ( 𝑤 = 𝑦 → ( 𝑥 ∧ 𝑤 ) = ( 𝑥 ∧ 𝑦 ) ) |
47 |
46
|
oveq2d |
⊢ ( 𝑤 = 𝑦 → ( 𝑧 ∨ ( 𝑥 ∧ 𝑤 ) ) = ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) |
48 |
|
oveq2 |
⊢ ( 𝑤 = 𝑦 → ( 𝑧 ∨ 𝑤 ) = ( 𝑧 ∨ 𝑦 ) ) |
49 |
48
|
oveq2d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) |
50 |
47 49
|
eqeq12d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑧 ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑤 ) ) ↔ ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) ) |
51 |
41 45 50
|
rspc3v |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) → ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) ) |
52 |
7 6 36 51
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) → ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) ) |
53 |
52
|
imp |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) = ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) |
54 |
53
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( 𝑥 ∧ ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) = ( 𝑥 ∧ ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) ) |
55 |
1 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑧 ∨ 𝑥 ) ∈ 𝐵 ) |
56 |
26 7 6 55
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑧 ∨ 𝑥 ) ∈ 𝐵 ) |
57 |
1 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∨ 𝑦 ) ∈ 𝐵 ) |
58 |
26 7 36 57
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑧 ∨ 𝑦 ) ∈ 𝐵 ) |
59 |
1 3
|
latmass |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑧 ∨ 𝑥 ) ∈ 𝐵 ∧ ( 𝑧 ∨ 𝑦 ) ∈ 𝐵 ) ) → ( ( 𝑥 ∧ ( 𝑧 ∨ 𝑥 ) ) ∧ ( 𝑧 ∨ 𝑦 ) ) = ( 𝑥 ∧ ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) ) |
60 |
26 6 56 58 59
|
syl13anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∧ ( 𝑧 ∨ 𝑥 ) ) ∧ ( 𝑧 ∨ 𝑦 ) ) = ( 𝑥 ∧ ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) ) |
61 |
1 2
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑧 ∨ 𝑥 ) = ( 𝑥 ∨ 𝑧 ) ) |
62 |
26 7 6 61
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑧 ∨ 𝑥 ) = ( 𝑥 ∨ 𝑧 ) ) |
63 |
62
|
oveq2d |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∧ ( 𝑧 ∨ 𝑥 ) ) = ( 𝑥 ∧ ( 𝑥 ∨ 𝑧 ) ) ) |
64 |
1 2 3
|
latabs2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) → ( 𝑥 ∧ ( 𝑥 ∨ 𝑧 ) ) = 𝑥 ) |
65 |
26 6 7 64
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∧ ( 𝑥 ∨ 𝑧 ) ) = 𝑥 ) |
66 |
63 65
|
eqtrd |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∧ ( 𝑧 ∨ 𝑥 ) ) = 𝑥 ) |
67 |
1 2
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∨ 𝑦 ) = ( 𝑦 ∨ 𝑧 ) ) |
68 |
26 7 36 67
|
syl3anc |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑧 ∨ 𝑦 ) = ( 𝑦 ∨ 𝑧 ) ) |
69 |
66 68
|
oveq12d |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 ∧ ( 𝑧 ∨ 𝑥 ) ) ∧ ( 𝑧 ∨ 𝑦 ) ) = ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) ) |
70 |
60 69
|
eqtr3d |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∧ ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) = ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) ) |
71 |
70
|
adantr |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( 𝑥 ∧ ( ( 𝑧 ∨ 𝑥 ) ∧ ( 𝑧 ∨ 𝑦 ) ) ) = ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) ) |
72 |
54 71
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( 𝑥 ∧ ( 𝑧 ∨ ( 𝑥 ∧ 𝑦 ) ) ) = ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) ) |
73 |
25 35 72
|
3eqtrrd |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) |
74 |
73
|
an32s |
⊢ ( ( ( 𝐾 ∈ Lat ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) |
75 |
74
|
ralrimivvva |
⊢ ( ( 𝐾 ∈ Lat ∧ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) |
76 |
75
|
ex |
⊢ ( 𝐾 ∈ Lat → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∨ ( 𝑣 ∧ 𝑤 ) ) = ( ( 𝑢 ∨ 𝑣 ) ∧ ( 𝑢 ∨ 𝑤 ) ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) ) |