Step |
Hyp |
Ref |
Expression |
1 |
|
pl42lem.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
pl42lem.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
pl42lem.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
pl42lem.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
pl42lem.o |
⊢ ⊥ = ( oc ‘ 𝐾 ) |
6 |
|
pl42lem.f |
⊢ 𝐹 = ( pmap ‘ 𝐾 ) |
7 |
|
pl42lem.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
8 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝐾 ∈ HL ) |
9 |
8
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝐾 ∈ Lat ) |
10 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
11 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
12 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
13 |
9 10 11 12
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) |
14 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
15 |
1 14 6
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
16 |
8 13 15
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
17 |
|
simpr2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑊 ∈ 𝐵 ) |
18 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑊 ) ∈ 𝐵 ) |
19 |
9 10 17 18
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝑋 ∨ 𝑊 ) ∈ 𝐵 ) |
20 |
|
simpr3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑉 ∈ 𝐵 ) |
21 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( 𝑌 ∨ 𝑉 ) ∈ 𝐵 ) |
22 |
9 11 20 21
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝑌 ∨ 𝑉 ) ∈ 𝐵 ) |
23 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∨ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∨ 𝑉 ) ∈ 𝐵 ) → ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ∈ 𝐵 ) |
24 |
9 19 22 23
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ∈ 𝐵 ) |
25 |
1 14 6
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
26 |
8 24 25
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
27 |
8 16 26
|
3jca |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ⊆ ( Atoms ‘ 𝐾 ) ) ) |
28 |
1 3 6 7
|
pmapjoin |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) |
29 |
9 10 11 28
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ) |
30 |
1 3 6 7
|
pmapjoin |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ) |
31 |
9 10 17 30
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ) |
32 |
1 3 6 7
|
pmapjoin |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) → ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ⊆ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) |
33 |
9 11 20 32
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ⊆ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) |
34 |
|
ss2in |
⊢ ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ∧ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ⊆ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ⊆ ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ∩ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) ) |
35 |
31 33 34
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ⊆ ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ∩ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) ) |
36 |
1 4 14 6
|
pmapmeet |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∨ 𝑊 ) ∈ 𝐵 ∧ ( 𝑌 ∨ 𝑉 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ∩ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) ) |
37 |
8 19 22 36
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) = ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑊 ) ) ∩ ( 𝐹 ‘ ( 𝑌 ∨ 𝑉 ) ) ) ) |
38 |
35 37
|
sseqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ⊆ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) |
39 |
29 38
|
jca |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∧ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ⊆ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ) |
40 |
14 7
|
paddss12 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) ∧ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ⊆ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ) ⊆ ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) + ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ) ) |
41 |
27 39 40
|
sylc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ) ⊆ ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) + ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ) |
42 |
1 3 6 7
|
pmapjoin |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ∧ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ∈ 𝐵 ) → ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) + ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ⊆ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ) |
43 |
9 13 24 42
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) + ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ⊆ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ) |
44 |
41 43
|
sstrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( 𝐹 ‘ 𝑌 ) + ( 𝐹 ‘ 𝑉 ) ) ) ) ⊆ ( 𝐹 ‘ ( ( 𝑋 ∨ 𝑌 ) ∨ ( ( 𝑋 ∨ 𝑊 ) ∧ ( 𝑌 ∨ 𝑉 ) ) ) ) ) |