Step |
Hyp |
Ref |
Expression |
1 |
|
2llnma1b.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
2llnma1b.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
2llnma1b.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
2llnma1b.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
2llnma1b.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝐾 ∈ Lat ) |
8 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ∈ 𝐴 ) |
9 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ∈ 𝐵 ) |
11 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
12 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑋 ) ) |
13 |
7 10 11 12
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑋 ) ) |
14 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑄 ∈ 𝐴 ) |
15 |
1 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
16 |
14 15
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑄 ∈ 𝐵 ) |
17 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
18 |
7 10 16 17
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
19 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ) |
20 |
7 10 11 19
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ) |
21 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝐾 ∈ HL ) |
22 |
1 3 5
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
23 |
21 8 14 22
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
24 |
1 2 4
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑋 ) ∧ 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) ) |
25 |
7 10 20 23 24
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑋 ) ∧ 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) ) |
26 |
13 18 25
|
mpbi2and |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) |
27 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
28 |
27
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝐾 ∈ AtLat ) |
29 |
|
simp3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) |
30 |
|
nbrne2 |
⊢ ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑋 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ≠ 𝑄 ) |
31 |
13 29 30
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ≠ 𝑄 ) |
32 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ∈ 𝐵 ) |
33 |
7 20 16 32
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ∈ 𝐵 ) |
34 |
1 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑋 ) ≤ ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ) |
35 |
7 20 16 34
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( 𝑃 ∨ 𝑋 ) ≤ ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ) |
36 |
1 2 7 10 20 33 13 35
|
lattrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ) |
37 |
1 2 3 4 5
|
cvrat3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ∧ 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |
38 |
37
|
3impia |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑃 ∨ 𝑋 ) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ∧ 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) |
39 |
21 20 8 14 31 29 36 38
|
syl133anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) |
40 |
2 5
|
atcmp |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) → ( 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ↔ 𝑃 = ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) ) |
41 |
28 8 39 40
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( 𝑃 ≤ ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ↔ 𝑃 = ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) ) |
42 |
26 41
|
mpbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → 𝑃 = ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ) |
43 |
42
|
eqcomd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑃 ∨ 𝑄 ) ) = 𝑃 ) |