Step |
Hyp |
Ref |
Expression |
1 |
|
2atjm.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
2atjm.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
2atjm.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
2atjm.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
2atjm.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ Lat ) |
8 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ∈ 𝐴 ) |
9 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ∈ 𝐵 ) |
11 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 ) |
12 |
1 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐵 ) |
14 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
15 |
7 10 13 14
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
16 |
|
simp3l |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≤ 𝑋 ) |
17 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
18 |
1 3 5
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
19 |
17 8 11 18
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
20 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
21 |
1 2 4
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≤ 𝑋 ) ↔ 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) ) |
22 |
7 10 19 20 21
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≤ 𝑋 ) ↔ 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) ) |
23 |
15 16 22
|
mpbi2and |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) |
24 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
25 |
24
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ AtLat ) |
26 |
1 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) = ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ) |
27 |
7 19 20 26
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) = ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ) |
28 |
20 8 11
|
3jca |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) |
29 |
|
nbrne2 |
⊢ ( ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) → 𝑃 ≠ 𝑄 ) |
30 |
29
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≠ 𝑄 ) |
31 |
|
simp3r |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ¬ 𝑄 ≤ 𝑋 ) |
32 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) |
33 |
7 20 13 32
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) |
34 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → 𝑋 ≤ ( 𝑋 ∨ 𝑄 ) ) |
35 |
7 20 13 34
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑋 ≤ ( 𝑋 ∨ 𝑄 ) ) |
36 |
1 2 7 10 20 33 16 35
|
lattrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) |
37 |
1 2 3 4 5
|
cvrat3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |
38 |
37
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) |
39 |
17 28 30 31 36 38
|
syl23anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) |
40 |
27 39
|
eqeltrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∈ 𝐴 ) |
41 |
2 5
|
atcmp |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∈ 𝐴 ) → ( 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ↔ 𝑃 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) ) |
42 |
25 8 40 41
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ↔ 𝑃 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) ) |
43 |
23 42
|
mpbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ) |
44 |
43
|
eqcomd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) = 𝑃 ) |