Step |
Hyp |
Ref |
Expression |
1 |
|
m.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
m.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
3 |
|
m.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
m.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
1 2 3 4
|
meetat |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑋 ∧ 𝑃 ) = 𝑃 ∨ ( 𝑋 ∧ 𝑃 ) = 0 ) ) |
6 |
|
eleq1a |
⊢ ( 𝑃 ∈ 𝐴 → ( ( 𝑋 ∧ 𝑃 ) = 𝑃 → ( 𝑋 ∧ 𝑃 ) ∈ 𝐴 ) ) |
7 |
6
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑋 ∧ 𝑃 ) = 𝑃 → ( 𝑋 ∧ 𝑃 ) ∈ 𝐴 ) ) |
8 |
7
|
orim1d |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( ( 𝑋 ∧ 𝑃 ) = 𝑃 ∨ ( 𝑋 ∧ 𝑃 ) = 0 ) → ( ( 𝑋 ∧ 𝑃 ) ∈ 𝐴 ∨ ( 𝑋 ∧ 𝑃 ) = 0 ) ) ) |
9 |
5 8
|
mpd |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑋 ∧ 𝑃 ) ∈ 𝐴 ∨ ( 𝑋 ∧ 𝑃 ) = 0 ) ) |