Step |
Hyp |
Ref |
Expression |
1 |
|
0ltat.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
2 |
|
0ltat.s |
⊢ < = ( lt ‘ 𝐾 ) |
3 |
|
0ltat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
simpl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ) → 𝐾 ∈ OP ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
6 |
5 1
|
op0cl |
⊢ ( 𝐾 ∈ OP → 0 ∈ ( Base ‘ 𝐾 ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ) → 0 ∈ ( Base ‘ 𝐾 ) ) |
8 |
5 3
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
10 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
11 |
1 10 3
|
atcvr0 |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ) → 0 ( ⋖ ‘ 𝐾 ) 𝑃 ) |
12 |
5 2 10
|
cvrlt |
⊢ ( ( ( 𝐾 ∈ OP ∧ 0 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) ∧ 0 ( ⋖ ‘ 𝐾 ) 𝑃 ) → 0 < 𝑃 ) |
13 |
4 7 9 11 12
|
syl31anc |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ) → 0 < 𝑃 ) |