Step |
Hyp |
Ref |
Expression |
1 |
|
cvrat3.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cvrat3.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cvrat3.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cvrat3.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cvrat3.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
7 |
1 2 3 6 5
|
cvr1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ) → ( ¬ 𝑄 ≤ 𝑋 ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ 𝑄 ) ) ) |
8 |
7
|
3adant3r2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ¬ 𝑄 ≤ 𝑋 ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ 𝑄 ) ) ) |
9 |
8
|
biimpa |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ ¬ 𝑄 ≤ 𝑋 ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ 𝑄 ) ) |
10 |
9
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ 𝑄 ) ) |
11 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
12 |
11
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ Lat ) |
13 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 ) |
14 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐵 ) |
16 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 ) |
17 |
1 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
18 |
16 17
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐵 ) |
19 |
1 3
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
20 |
12 15 18 19
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
21 |
20
|
oveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( 𝑋 ∨ ( 𝑄 ∨ 𝑃 ) ) ) |
22 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑋 ∈ 𝐵 ) |
23 |
1 3
|
latjass |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) = ( 𝑋 ∨ ( 𝑄 ∨ 𝑃 ) ) ) |
24 |
12 22 18 15 23
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) = ( 𝑋 ∨ ( 𝑄 ∨ 𝑃 ) ) ) |
25 |
21 24
|
eqtr4d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) ) |
26 |
25
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) ) |
27 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) |
28 |
12 22 18 27
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) |
29 |
1 2 3
|
latjlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) ) → ( 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) → ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) ≤ ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) ) ) |
30 |
12 15 28 28 29
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) → ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) ≤ ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) ) ) |
31 |
30
|
imp |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( ( 𝑋 ∨ 𝑄 ) ∨ 𝑃 ) ≤ ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) ) |
32 |
26 31
|
eqbrtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ≤ ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) ) |
33 |
1 3
|
latjidm |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) → ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) |
34 |
12 28 33
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) |
35 |
34
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( ( 𝑋 ∨ 𝑄 ) ∨ ( 𝑋 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) |
36 |
32 35
|
breqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ≤ ( 𝑋 ∨ 𝑄 ) ) |
37 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ HL ) |
38 |
2 3 5
|
hlatlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) ) |
39 |
37 13 16 38
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) ) |
40 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
41 |
12 15 18 40
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) |
42 |
1 2 3
|
latjlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ) |
43 |
12 18 41 22 42
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ) |
44 |
39 43
|
mpd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
45 |
44
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
46 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ) |
47 |
12 22 41 46
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ) |
48 |
1 2
|
latasymb |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑄 ) ∈ 𝐵 ) → ( ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ≤ ( 𝑋 ∨ 𝑄 ) ∧ ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ↔ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) ) |
49 |
12 47 28 48
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ≤ ( 𝑋 ∨ 𝑄 ) ∧ ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ↔ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) ) |
50 |
49
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( ( ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ≤ ( 𝑋 ∨ 𝑄 ) ∧ ( 𝑋 ∨ 𝑄 ) ≤ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ↔ ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) ) |
51 |
36 45 50
|
mpbi2and |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( 𝑋 ∨ 𝑄 ) ) |
52 |
51
|
breq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ 𝑄 ) ) ) |
53 |
52
|
adantrl |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) ) → ( 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ 𝑄 ) ) ) |
54 |
10 53
|
mpbird |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
55 |
54
|
ex |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ) |
56 |
1 3 4 6
|
cvrexch |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ) |
57 |
37 22 41 56
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ↔ 𝑋 ( ⋖ ‘ 𝐾 ) ( 𝑋 ∨ ( 𝑃 ∨ 𝑄 ) ) ) ) |
58 |
55 57
|
sylibrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) |
59 |
58
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) |
60 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑄 ) ∈ 𝐵 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ) |
61 |
12 22 41 60
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ) |
62 |
1 3 6 5
|
cvrat2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) |
63 |
62
|
3expia |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |
64 |
37 61 13 16 63
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |
65 |
64
|
expdimp |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |
66 |
59 65
|
syld |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑃 ≠ 𝑄 ) → ( ( ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |
67 |
66
|
exp4b |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ≠ 𝑄 → ( ¬ 𝑄 ≤ 𝑋 → ( 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) ) ) |
68 |
67
|
3impd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ ( 𝑋 ∨ 𝑄 ) ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 ) ) |