Step |
Hyp |
Ref |
Expression |
1 |
|
latdisd.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
latdisd.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
latdisd.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
1 2 3
|
latdisdlem |
⊢ ( 𝐾 ∈ Lat → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ∨ ( 𝑦 ∧ 𝑧 ) ) = ( ( 𝑥 ∨ 𝑦 ) ∧ ( 𝑥 ∨ 𝑧 ) ) → ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( ( 𝑢 ∧ 𝑣 ) ∨ ( 𝑢 ∧ 𝑤 ) ) ) ) |
5 |
|
eqid |
⊢ ( ODual ‘ 𝐾 ) = ( ODual ‘ 𝐾 ) |
6 |
5
|
odulat |
⊢ ( 𝐾 ∈ Lat → ( ODual ‘ 𝐾 ) ∈ Lat ) |
7 |
5 1
|
odubas |
⊢ 𝐵 = ( Base ‘ ( ODual ‘ 𝐾 ) ) |
8 |
5 3
|
odujoin |
⊢ ∧ = ( join ‘ ( ODual ‘ 𝐾 ) ) |
9 |
5 2
|
odumeet |
⊢ ∨ = ( meet ‘ ( ODual ‘ 𝐾 ) ) |
10 |
7 8 9
|
latdisdlem |
⊢ ( ( ODual ‘ 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 ) ) ) ) |
11 |
6 10
|
syl |
⊢ ( 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 ) ) ) ) |
12 |
4 11
|
impbid |
⊢ ( 𝐾 ∈ Lat → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ∨ ( 𝑦 ∧ 𝑧 ) ) = ( ( 𝑥 ∨ 𝑦 ) ∧ ( 𝑥 ∨ 𝑧 ) ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ∀ 𝑤 ∈ 𝐵 ( 𝑢 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( ( 𝑢 ∧ 𝑣 ) ∨ ( 𝑢 ∧ 𝑤 ) ) ) ) |
13 |
|
oveq1 |
⊢ ( 𝑢 = 𝑥 → ( 𝑢 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( 𝑥 ∧ ( 𝑣 ∨ 𝑤 ) ) ) |
14 |
|
oveq1 |
⊢ ( 𝑢 = 𝑥 → ( 𝑢 ∧ 𝑣 ) = ( 𝑥 ∧ 𝑣 ) ) |
15 |
|
oveq1 |
⊢ ( 𝑢 = 𝑥 → ( 𝑢 ∧ 𝑤 ) = ( 𝑥 ∧ 𝑤 ) ) |
16 |
14 15
|
oveq12d |
⊢ ( 𝑢 = 𝑥 → ( ( 𝑢 ∧ 𝑣 ) ∨ ( 𝑢 ∧ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑣 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ) |
17 |
13 16
|
eqeq12d |
⊢ ( 𝑢 = 𝑥 → ( ( 𝑢 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( ( 𝑢 ∧ 𝑣 ) ∨ ( 𝑢 ∧ 𝑤 ) ) ↔ ( 𝑥 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑣 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ) ) |
18 |
|
oveq1 |
⊢ ( 𝑣 = 𝑦 → ( 𝑣 ∨ 𝑤 ) = ( 𝑦 ∨ 𝑤 ) ) |
19 |
18
|
oveq2d |
⊢ ( 𝑣 = 𝑦 → ( 𝑥 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( 𝑥 ∧ ( 𝑦 ∨ 𝑤 ) ) ) |
20 |
|
oveq2 |
⊢ ( 𝑣 = 𝑦 → ( 𝑥 ∧ 𝑣 ) = ( 𝑥 ∧ 𝑦 ) ) |
21 |
20
|
oveq1d |
⊢ ( 𝑣 = 𝑦 → ( ( 𝑥 ∧ 𝑣 ) ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ) |
22 |
19 21
|
eqeq12d |
⊢ ( 𝑣 = 𝑦 → ( ( 𝑥 ∧ ( 𝑣 ∨ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑣 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ↔ ( 𝑥 ∧ ( 𝑦 ∨ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ) ) |
23 |
|
oveq2 |
⊢ ( 𝑤 = 𝑧 → ( 𝑦 ∨ 𝑤 ) = ( 𝑦 ∨ 𝑧 ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝑤 = 𝑧 → ( 𝑥 ∧ ( 𝑦 ∨ 𝑤 ) ) = ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑤 = 𝑧 → ( 𝑥 ∧ 𝑤 ) = ( 𝑥 ∧ 𝑧 ) ) |
26 |
25
|
oveq2d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) |
27 |
24 26
|
eqeq12d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑥 ∧ ( 𝑦 ∨ 𝑤 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑤 ) ) ↔ ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) ) |
28 |
17 22 27
|
cbvral3vw |
⊢ ( ∀ 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 ) ) ) |
29 |
12 28
|
bitrdi |
⊢ ( 𝐾 ∈ Lat → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ∨ ( 𝑦 ∧ 𝑧 ) ) = ( ( 𝑥 ∨ 𝑦 ) ∧ ( 𝑥 ∨ 𝑧 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑥 ∧ ( 𝑦 ∨ 𝑧 ) ) = ( ( 𝑥 ∧ 𝑦 ) ∨ ( 𝑥 ∧ 𝑧 ) ) ) ) |