Step |
Hyp |
Ref |
Expression |
1 |
|
atcvreq0.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atcvreq0.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atcvreq0.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
4 |
|
atcvreq0.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
5 |
|
atcvreq0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
7 |
1 6 3
|
atl0le |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ) → 0 ( le ‘ 𝐾 ) 𝑋 ) |
8 |
7
|
3adant3 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 0 ( le ‘ 𝐾 ) 𝑋 ) |
9 |
8
|
adantr |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 0 ( le ‘ 𝐾 ) 𝑋 ) |
10 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
11 |
|
eqid |
⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 ) |
12 |
1 11 4
|
cvrlt |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑃 ) → 𝑋 ( lt ‘ 𝐾 ) 𝑃 ) |
13 |
10 12
|
syl3anl3 |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 𝑋 ( lt ‘ 𝐾 ) 𝑃 ) |
14 |
|
atlpos |
⊢ ( 𝐾 ∈ AtLat → 𝐾 ∈ Poset ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 𝐾 ∈ Poset ) |
16 |
15
|
adantr |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 𝐾 ∈ Poset ) |
17 |
1 3
|
atl0cl |
⊢ ( 𝐾 ∈ AtLat → 0 ∈ 𝐵 ) |
18 |
17
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 0 ∈ 𝐵 ) |
19 |
18
|
adantr |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 0 ∈ 𝐵 ) |
20 |
10
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ 𝐵 ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 𝑃 ∈ 𝐵 ) |
22 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 𝑋 ∈ 𝐵 ) |
23 |
3 4 5
|
atcvr0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → 0 𝐶 𝑃 ) |
24 |
23
|
3adant2 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → 0 𝐶 𝑃 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 0 𝐶 𝑃 ) |
26 |
1 6 11 4
|
cvrnbtwn3 |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 0 𝐶 𝑃 ) → ( ( 0 ( le ‘ 𝐾 ) 𝑋 ∧ 𝑋 ( lt ‘ 𝐾 ) 𝑃 ) ↔ 0 = 𝑋 ) ) |
27 |
16 19 21 22 25 26
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → ( ( 0 ( le ‘ 𝐾 ) 𝑋 ∧ 𝑋 ( lt ‘ 𝐾 ) 𝑃 ) ↔ 0 = 𝑋 ) ) |
28 |
9 13 27
|
mpbi2and |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 0 = 𝑋 ) |
29 |
28
|
eqcomd |
⊢ ( ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑋 𝐶 𝑃 ) → 𝑋 = 0 ) |
30 |
29
|
ex |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 𝐶 𝑃 → 𝑋 = 0 ) ) |
31 |
|
breq1 |
⊢ ( 𝑋 = 0 → ( 𝑋 𝐶 𝑃 ↔ 0 𝐶 𝑃 ) ) |
32 |
24 31
|
syl5ibrcom |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 = 0 → 𝑋 𝐶 𝑃 ) ) |
33 |
30 32
|
impbid |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑋 𝐶 𝑃 ↔ 𝑋 = 0 ) ) |