Step |
Hyp |
Ref |
Expression |
1 |
|
atmod.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atmod.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atmod.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
atmod.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
atmod.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝐾 ∈ HL ) |
7 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑃 ∈ 𝐴 ) |
8 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑌 ∈ 𝐵 ) |
9 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑋 ∈ 𝐵 ) |
10 |
|
simp3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑃 ≤ 𝑋 ) |
11 |
1 2 3 4 5
|
atmod1i1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑃 ∨ ( 𝑌 ∧ 𝑋 ) ) = ( ( 𝑃 ∨ 𝑌 ) ∧ 𝑋 ) ) |
12 |
6 7 8 9 10 11
|
syl131anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑃 ∨ ( 𝑌 ∧ 𝑋 ) ) = ( ( 𝑃 ∨ 𝑌 ) ∧ 𝑋 ) ) |
13 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
14 |
13
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝐾 ∈ Lat ) |
15 |
1 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑋 ) ) |
16 |
14 9 8 15
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑋 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑋 ) ) |
17 |
16
|
oveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑃 ∨ ( 𝑋 ∧ 𝑌 ) ) = ( 𝑃 ∨ ( 𝑌 ∧ 𝑋 ) ) ) |
18 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
19 |
7 18
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → 𝑃 ∈ 𝐵 ) |
20 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ∨ 𝑌 ) ∈ 𝐵 ) |
21 |
14 19 8 20
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑃 ∨ 𝑌 ) ∈ 𝐵 ) |
22 |
1 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑃 ∨ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑌 ) ) = ( ( 𝑃 ∨ 𝑌 ) ∧ 𝑋 ) ) |
23 |
14 9 21 22
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑋 ∧ ( 𝑃 ∨ 𝑌 ) ) = ( ( 𝑃 ∨ 𝑌 ) ∧ 𝑋 ) ) |
24 |
12 17 23
|
3eqtr4d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑃 ≤ 𝑋 ) → ( 𝑃 ∨ ( 𝑋 ∧ 𝑌 ) ) = ( 𝑋 ∧ ( 𝑃 ∨ 𝑌 ) ) ) |