Step |
Hyp |
Ref |
Expression |
1 |
|
cvlatcvr1.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
cvlatcvr1.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
3 |
|
cvlatcvr1.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
simp13 |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ CvLat ) |
5 |
|
cvlatl |
⊢ ( 𝐾 ∈ CvLat → 𝐾 ∈ AtLat ) |
6 |
4 5
|
syl |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ AtLat ) |
7 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
9 |
7 8 3
|
atnem0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ) ) |
10 |
6 9
|
syld3an1 |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
12 |
11 3
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
13 |
11 1 7 8 2 3
|
cvlcvrp |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) |
14 |
12 13
|
syl3an2 |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ( meet ‘ 𝐾 ) 𝑄 ) = ( 0. ‘ 𝐾 ) ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) |
15 |
10 14
|
bitrd |
⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ CvLat ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ 𝑃 𝐶 ( 𝑃 ∨ 𝑄 ) ) ) |