Step |
Hyp |
Ref |
Expression |
1 |
|
1cvrat.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
1cvrat.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
1cvrat.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
1cvrat.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
1cvrat.u |
⊢ 1 = ( 1. ‘ 𝐾 ) |
6 |
|
1cvrat.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
7 |
|
1cvrat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
8 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
9 |
8
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝐾 ∈ Lat ) |
10 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑃 ∈ 𝐴 ) |
11 |
1 7
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑃 ∈ 𝐵 ) |
13 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 ) |
14 |
1 7
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐵 ) |
16 |
1 3
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
17 |
9 12 15 16
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
18 |
17
|
oveq1d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑋 ) ) |
19 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑄 ∨ 𝑃 ) ∈ 𝐵 ) |
20 |
9 15 12 19
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( 𝑄 ∨ 𝑃 ) ∈ 𝐵 ) |
21 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
22 |
1 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑃 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑋 ) = ( 𝑋 ∧ ( 𝑄 ∨ 𝑃 ) ) ) |
23 |
9 20 21 22
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑋 ) = ( 𝑋 ∧ ( 𝑄 ∨ 𝑃 ) ) ) |
24 |
18 23
|
eqtrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) = ( 𝑋 ∧ ( 𝑄 ∨ 𝑃 ) ) ) |
25 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
26 |
21 13 10
|
3jca |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ) |
27 |
|
simp31 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑃 ≠ 𝑄 ) |
28 |
27
|
necomd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑄 ≠ 𝑃 ) |
29 |
|
simp33 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ¬ 𝑃 ≤ 𝑋 ) |
30 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
31 |
30
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝐾 ∈ OP ) |
32 |
1 2 5
|
ople1 |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑄 ∈ 𝐵 ) → 𝑄 ≤ 1 ) |
33 |
31 15 32
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑄 ≤ 1 ) |
34 |
|
simp32 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑋 𝐶 1 ) |
35 |
1 2 3 5 6 7
|
1cvrjat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( 𝑋 ∨ 𝑃 ) = 1 ) |
36 |
25 21 10 34 29 35
|
syl32anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( 𝑋 ∨ 𝑃 ) = 1 ) |
37 |
33 36
|
breqtrrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → 𝑄 ≤ ( 𝑋 ∨ 𝑃 ) ) |
38 |
1 2 3 4 7
|
cvrat3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ) → ( ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ ( 𝑋 ∨ 𝑃 ) ) → ( 𝑋 ∧ ( 𝑄 ∨ 𝑃 ) ) ∈ 𝐴 ) ) |
39 |
38
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ) ∧ ( 𝑄 ≠ 𝑃 ∧ ¬ 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ ( 𝑋 ∨ 𝑃 ) ) ) → ( 𝑋 ∧ ( 𝑄 ∨ 𝑃 ) ) ∈ 𝐴 ) |
40 |
25 26 28 29 37 39
|
syl23anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑄 ∨ 𝑃 ) ) ∈ 𝐴 ) |
41 |
24 40
|
eqeltrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑋 𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑋 ) ∈ 𝐴 ) |