Step |
Hyp |
Ref |
Expression |
1 |
|
leatom.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
leatom.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
leatom.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
leatom.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
1 2 3 4
|
leat |
⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) ) |
6 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → 𝑃 ∈ 𝐴 ) |
7 |
|
eleq1a |
⊢ ( 𝑃 ∈ 𝐴 → ( 𝑋 = 𝑃 → 𝑋 ∈ 𝐴 ) ) |
8 |
6 7
|
syl |
⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 = 𝑃 → 𝑋 ∈ 𝐴 ) ) |
9 |
8
|
orim1d |
⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( ( 𝑋 = 𝑃 ∨ 𝑋 = 0 ) → ( 𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ) ) ) |
10 |
5 9
|
mpd |
⊢ ( ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 ≤ 𝑃 ) → ( 𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ) ) |