Step |
Hyp |
Ref |
Expression |
1 |
|
atlen0.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atlen0.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atlen0.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
atlen0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝐾 ∈ AtLat ) |
6 |
1 3
|
atl0cl |
⊢ ( 𝐾 ∈ AtLat → 0 ∈ 𝐵 ) |
7 |
5 6
|
syl |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 0 ∈ 𝐵 ) |
8 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑋 ∈ 𝐵 ) |
9 |
5 7 8
|
3jca |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) |
10 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑃 ∈ 𝐴 ) |
11 |
1 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
12 |
10 11
|
syl |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑃 ∈ 𝐵 ) |
13 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
14 |
3 13 4
|
atcvr0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → 0 ( ⋖ ‘ 𝐾 ) 𝑃 ) |
15 |
5 10 14
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 0 ( ⋖ ‘ 𝐾 ) 𝑃 ) |
16 |
|
eqid |
⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 ) |
17 |
1 16 13
|
cvrlt |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) ∧ 0 ( ⋖ ‘ 𝐾 ) 𝑃 ) → 0 ( lt ‘ 𝐾 ) 𝑃 ) |
18 |
5 7 12 15 17
|
syl31anc |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 0 ( lt ‘ 𝐾 ) 𝑃 ) |
19 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑃 ≤ 𝑋 ) |
20 |
|
atlpos |
⊢ ( 𝐾 ∈ AtLat → 𝐾 ∈ Poset ) |
21 |
5 20
|
syl |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝐾 ∈ Poset ) |
22 |
1 2 16
|
pltletr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 0 ( lt ‘ 𝐾 ) 𝑃 ∧ 𝑃 ≤ 𝑋 ) → 0 ( lt ‘ 𝐾 ) 𝑋 ) ) |
23 |
21 7 12 8 22
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → ( ( 0 ( lt ‘ 𝐾 ) 𝑃 ∧ 𝑃 ≤ 𝑋 ) → 0 ( lt ‘ 𝐾 ) 𝑋 ) ) |
24 |
18 19 23
|
mp2and |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 0 ( lt ‘ 𝐾 ) 𝑋 ) |
25 |
16
|
pltne |
⊢ ( ( 𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 0 ( lt ‘ 𝐾 ) 𝑋 → 0 ≠ 𝑋 ) ) |
26 |
9 24 25
|
sylc |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 0 ≠ 𝑋 ) |
27 |
26
|
necomd |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑋 ≠ 0 ) |