Step |
Hyp |
Ref |
Expression |
1 |
|
hlatjcom.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
hlatjcom.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
4 |
3
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝐾 ∈ Lat ) |
5 |
|
simp2l |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
6 2
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
8 |
5 7
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
9 |
|
simp2r |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 ) |
10 |
6 2
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
11 |
9 10
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
12 |
|
simp3l |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 ) |
13 |
6 2
|
atbase |
⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
15 |
|
simp3r |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑆 ∈ 𝐴 ) |
16 |
6 2
|
atbase |
⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
17 |
15 16
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
18 |
6 1
|
latj4 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑅 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑅 ) ∨ ( 𝑄 ∨ 𝑆 ) ) ) |
19 |
4 8 11 14 17 18
|
syl122anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑅 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑅 ) ∨ ( 𝑄 ∨ 𝑆 ) ) ) |