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 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝐾 ∈ Lat ) |
8 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑌 ∈ 𝐵 ) |
9 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑋 ∈ 𝐵 ) |
10 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑃 ∈ 𝐴 ) |
11 |
1 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵 ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → 𝑃 ∈ 𝐵 ) |
13 |
1 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 ) |
14 |
7 9 12 13
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 ) |
15 |
1 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∨ 𝑃 ) ∈ 𝐵 ) → ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) ) |
16 |
7 8 14 15
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) ) |
17 |
1 2 3 4 5
|
atmod1i2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑃 ∧ 𝑌 ) ) = ( ( 𝑋 ∨ 𝑃 ) ∧ 𝑌 ) ) |
18 |
1 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑃 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑃 ) ) |
19 |
7 12 8 18
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑃 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑃 ) ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑃 ∧ 𝑌 ) ) = ( 𝑋 ∨ ( 𝑌 ∧ 𝑃 ) ) ) |
21 |
16 17 20
|
3eqtr2rd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ∨ ( 𝑌 ∧ 𝑃 ) ) = ( 𝑌 ∧ ( 𝑋 ∨ 𝑃 ) ) ) |