Step |
Hyp |
Ref |
Expression |
1 |
|
atcmp.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
atcmp.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
atlpos |
⊢ ( 𝐾 ∈ AtLat → 𝐾 ∈ Poset ) |
4 |
3
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ Poset ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
6 |
5 2
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
7 |
6
|
3ad2ant2 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
8 |
5 2
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
9 |
8
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
10 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
11 |
5 10
|
atl0cl |
⊢ ( 𝐾 ∈ AtLat → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
12 |
11
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
14 |
10 13 2
|
atcvr0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑃 ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑃 ) |
16 |
10 13 2
|
atcvr0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑄 ) |
17 |
16
|
3adant2 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑄 ) |
18 |
5 1 13
|
cvrcmp |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 0. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑃 ∧ ( 0. ‘ 𝐾 ) ( ⋖ ‘ 𝐾 ) 𝑄 ) ) → ( 𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄 ) ) |
19 |
4 7 9 12 15 17 18
|
syl132anc |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄 ) ) |