Step |
Hyp |
Ref |
Expression |
1 |
|
lhpmod.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lhpmod.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
lhpmod.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
lhpmod.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
lhpmod.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝐾 ∈ HL ) |
7 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑊 ∈ 𝐻 ) |
8 |
|
eqid |
⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
10 |
8 9 5
|
lhpocat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ) |
11 |
6 7 10
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ) |
12 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
13 |
6 12
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝐾 ∈ OP ) |
14 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑋 ∈ 𝐵 ) |
15 |
1 8
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
16 |
13 14 15
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ) |
17 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑌 ∈ 𝐵 ) |
18 |
1 8
|
opoccl |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) |
19 |
13 17 18
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) |
20 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑌 ≤ 𝑋 ) |
21 |
1 2 8
|
oplecon3b |
⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑋 ↔ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ≤ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
22 |
13 17 14 21
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑌 ≤ 𝑋 ↔ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ≤ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
23 |
20 22
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ≤ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) |
24 |
1 2 3 4 9
|
atmod1i2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝐵 ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ∈ 𝐵 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ≤ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
25 |
6 11 16 19 23 24
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
26 |
6
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝐾 ∈ Lat ) |
27 |
1 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
28 |
7 27
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝑊 ∈ 𝐵 ) |
29 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
30 |
26 14 28 29
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
31 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ∈ 𝐵 ) |
32 |
26 30 17 31
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ∈ 𝐵 ) |
33 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑊 ∨ 𝑌 ) ∈ 𝐵 ) |
34 |
26 28 17 33
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑊 ∨ 𝑌 ) ∈ 𝐵 ) |
35 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑊 ∨ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ∈ 𝐵 ) |
36 |
26 14 34 35
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ∈ 𝐵 ) |
37 |
1 8
|
opcon3b |
⊢ ( ( 𝐾 ∈ OP ∧ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ∈ 𝐵 ∧ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ∈ 𝐵 ) → ( ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) = ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ↔ ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ) ) ) |
38 |
13 32 36 37
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) = ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ↔ ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ) ) ) |
39 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
40 |
6 39
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → 𝐾 ∈ OL ) |
41 |
1 3 4 8
|
oldmm1 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑊 ∨ 𝑌 ) ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ ( 𝑊 ∨ 𝑌 ) ) ) ) |
42 |
40 14 34 41
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ ( 𝑊 ∨ 𝑌 ) ) ) ) |
43 |
1 3 4 8
|
oldmj1 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑊 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑊 ∨ 𝑌 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
44 |
40 28 17 43
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑊 ∨ 𝑌 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
45 |
44
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
46 |
42 45
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
47 |
1 3 4 8
|
oldmj1 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
48 |
40 30 17 47
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
49 |
1 3 4 8
|
oldmm1 |
⊢ ( ( 𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
50 |
40 14 28 49
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ 𝑊 ) ) = ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) |
51 |
50
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
52 |
48 51
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) |
53 |
46 52
|
eqeq12d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( ( oc ‘ 𝐾 ) ‘ ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) = ( ( oc ‘ 𝐾 ) ‘ ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) ) ↔ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
54 |
38 53
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) = ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ↔ ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑋 ) ∨ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑌 ) ) ) ) |
55 |
25 54
|
mpbird |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑌 ≤ 𝑋 ) → ( ( 𝑋 ∧ 𝑊 ) ∨ 𝑌 ) = ( 𝑋 ∧ ( 𝑊 ∨ 𝑌 ) ) ) |