Step |
Hyp |
Ref |
Expression |
1 |
|
atncvr.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
2 |
|
atncvr.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
4 |
3 2
|
atn0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) ) |
5 |
4
|
3adant3 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≠ ( 0. ‘ 𝐾 ) ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
6 2
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
8 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
9 |
6 8 3 1 2
|
atcvreq0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 𝐶 𝑄 ↔ 𝑃 = ( 0. ‘ 𝐾 ) ) ) |
10 |
7 9
|
syl3an2 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 𝐶 𝑄 ↔ 𝑃 = ( 0. ‘ 𝐾 ) ) ) |
11 |
10
|
necon3bbid |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ¬ 𝑃 𝐶 𝑄 ↔ 𝑃 ≠ ( 0. ‘ 𝐾 ) ) ) |
12 |
5 11
|
mpbird |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 𝐶 𝑄 ) |