Step |
Hyp |
Ref |
Expression |
1 |
|
hlrelat3.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
hlrelat3.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
hlrelat3.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
|
hlrelat3.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
hlrelat3.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
6 |
|
hlrelat3.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
1 2 3 6
|
hlrelat1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) ) |
8 |
7
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) |
9 |
|
simp3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → ¬ 𝑝 ≤ 𝑋 ) |
10 |
|
simp1l1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ HL ) |
11 |
|
simp1l2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
12 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑝 ∈ 𝐴 ) |
13 |
1 2 4 5 6
|
cvr1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐴 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ) ) |
14 |
10 11 12 13
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ) ) |
15 |
9 14
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ) |
16 |
|
simp1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) |
17 |
|
simp1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑋 < 𝑌 ) |
18 |
2 3
|
pltle |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → 𝑋 ≤ 𝑌 ) ) |
19 |
16 17 18
|
sylc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑋 ≤ 𝑌 ) |
20 |
|
simp3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑝 ≤ 𝑌 ) |
21 |
10
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ Lat ) |
22 |
1 6
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 ) |
23 |
12 22
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑝 ∈ 𝐵 ) |
24 |
|
simp1l3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
25 |
1 2 4
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |
26 |
21 11 23 24 25
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |
27 |
19 20 26
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) |
28 |
15 27
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) → ( 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |
29 |
28
|
3exp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ( 𝑝 ∈ 𝐴 → ( ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) → ( 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) ) ) |
30 |
29
|
reximdvai |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) ) |
31 |
8 30
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑋 𝐶 ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |