Step |
Hyp |
Ref |
Expression |
1 |
|
atlelt.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atlelt.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atlelt.s |
⊢ < = ( lt ‘ 𝐾 ) |
4 |
|
atlelt.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
simp3r |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 < 𝑋 ) |
6 |
|
breq1 |
⊢ ( 𝑃 = 𝑄 → ( 𝑃 < 𝑋 ↔ 𝑄 < 𝑋 ) ) |
7 |
5 6
|
syl5ibrcom |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 = 𝑄 → 𝑃 < 𝑋 ) ) |
8 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝐾 ∈ HL ) |
9 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 ∈ 𝐴 ) |
10 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 ∈ 𝐴 ) |
11 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
12 |
3 11 4
|
atlt |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) ) |
13 |
8 9 10 12
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) ) |
14 |
|
simp3l |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 ≤ 𝑋 ) |
15 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
16 |
8 10 15
|
3jca |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ) |
17 |
2 3
|
pltle |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑄 < 𝑋 → 𝑄 ≤ 𝑋 ) ) |
18 |
16 5 17
|
sylc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 ≤ 𝑋 ) |
19 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
20 |
19
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝐾 ∈ Lat ) |
21 |
1 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
22 |
9 21
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 ∈ 𝐵 ) |
23 |
1 4
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
24 |
10 23
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑄 ∈ 𝐵 ) |
25 |
1 2 11
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋 ) ↔ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) ) |
26 |
20 22 24 15 25
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( ( 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋 ) ↔ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) ) |
27 |
14 18 26
|
mpbi2and |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) |
28 |
|
hlpos |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Poset ) |
29 |
28
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝐾 ∈ Poset ) |
30 |
1 11
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 ) |
31 |
20 22 24 30
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 ) |
32 |
1 2 3
|
pltletr |
⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑃 ∈ 𝐵 ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) → 𝑃 < 𝑋 ) ) |
33 |
29 22 31 15 32
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ∧ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ≤ 𝑋 ) → 𝑃 < 𝑋 ) ) |
34 |
27 33
|
mpan2d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 < ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) → 𝑃 < 𝑋 ) ) |
35 |
13 34
|
sylbird |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → ( 𝑃 ≠ 𝑄 → 𝑃 < 𝑋 ) ) |
36 |
7 35
|
pm2.61dne |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ 𝑄 < 𝑋 ) ) → 𝑃 < 𝑋 ) |