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 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
10 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
11 |
1 10 6
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
12 |
8 9 11
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
13 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 ) |
14 |
1 10 6
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
15 |
8 13 14
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
16 |
10 7
|
paddssat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑌 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
17 |
8 12 15 16
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) |
18 |
|
simpr2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑊 ∈ 𝐵 ) |
19 |
1 10 6
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑊 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
20 |
8 18 19
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑊 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
21 |
|
inss1 |
⊢ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) ⊆ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) |
22 |
10 7
|
paddss1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑊 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) ⊆ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) → ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑊 ) ) ) ) |
23 |
21 22
|
mpi |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑊 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑊 ) ) ) |
24 |
8 17 20 23
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑊 ) ) ) |
25 |
|
simpr3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → 𝑉 ∈ 𝐵 ) |
26 |
1 10 6
|
pmapssat |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑉 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑉 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
27 |
8 25 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑉 ) ⊆ ( Atoms ‘ 𝐾 ) ) |
28 |
10 7
|
sspadd2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑉 ) ⊆ ( Atoms ‘ 𝐾 ) ∧ ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑉 ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑉 ) ) ) |
29 |
8 27 17 28
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( 𝐹 ‘ 𝑉 ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑉 ) ) ) |
30 |
|
ss2in |
⊢ ( ( ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) + ( 𝐹 ‘ 𝑊 ) ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑉 ) ⊆ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑉 ) ) ) → ( ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( 𝐹 ‘ 𝑉 ) ) ⊆ ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑉 ) ) ) ) |
31 |
24 29 30
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵 ) ) → ( ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) ∩ ( 𝐹 ‘ 𝑍 ) ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( 𝐹 ‘ 𝑉 ) ) ⊆ ( ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑊 ) ) ∩ ( ( ( 𝐹 ‘ 𝑋 ) + ( 𝐹 ‘ 𝑌 ) ) + ( 𝐹 ‘ 𝑉 ) ) ) ) |