Step |
Hyp |
Ref |
Expression |
1 |
|
hlatjcom.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
hlatjcom.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
1 2
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
4 |
3
|
3adant3r3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) ) |
5 |
4
|
oveq1d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( ( 𝑄 ∨ 𝑃 ) ∨ 𝑅 ) ) |
6 |
1 2
|
hlatjass |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) ) |
7 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ HL ) |
8 |
|
simpr2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 ) |
9 |
|
simpr1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 ) |
10 |
|
simpr3 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 ) |
11 |
1 2
|
hlatjass |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑄 ∨ 𝑃 ) ∨ 𝑅 ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑅 ) ) ) |
12 |
7 8 9 10 11
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑄 ∨ 𝑃 ) ∨ 𝑅 ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑅 ) ) ) |
13 |
5 6 12
|
3eqtr3d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑅 ) ) ) |