Step |
Hyp |
Ref |
Expression |
1 |
|
hlrelat5.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
hlrelat5.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
hlrelat5.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
|
hlrelat5.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
hlrelat5.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
1 2 3 5
|
hlrelat1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) ) |
7 |
6
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) |
8 |
|
simpll1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ HL ) |
9 |
8
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝐾 ∈ Lat ) |
10 |
|
simpll2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 ) |
11 |
1 5
|
atbase |
⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 ) |
12 |
11
|
adantl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑝 ∈ 𝐵 ) |
13 |
1 2 3 4
|
latnle |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑝 ) ) ) |
14 |
9 10 12 13
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( ¬ 𝑝 ≤ 𝑋 ↔ 𝑋 < ( 𝑋 ∨ 𝑝 ) ) ) |
15 |
2 3
|
pltle |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → 𝑋 ≤ 𝑌 ) ) |
16 |
15
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → 𝑋 ≤ 𝑌 ) |
17 |
16
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑋 ≤ 𝑌 ) |
18 |
17
|
biantrurd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ) ) |
19 |
|
simpll3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → 𝑌 ∈ 𝐵 ) |
20 |
1 2 4
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |
21 |
9 10 12 19 20
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |
22 |
18 21
|
bitrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( 𝑝 ≤ 𝑌 ↔ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |
23 |
14 22
|
anbi12d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ↔ ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) ) |
24 |
23
|
rexbidva |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ↔ ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) ) |
25 |
7 24
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 < 𝑌 ) → ∃ 𝑝 ∈ 𝐴 ( 𝑋 < ( 𝑋 ∨ 𝑝 ) ∧ ( 𝑋 ∨ 𝑝 ) ≤ 𝑌 ) ) |