Step |
Hyp |
Ref |
Expression |
1 |
|
hlatexch4.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
hlatexch4.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
simp1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝐾 ∈ HL ) |
4 |
|
simp21 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑃 ∈ 𝐴 ) |
5 |
|
simp22 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ∈ 𝐴 ) |
6 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
7 |
6 1 2
|
hlatlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) |
8 |
3 4 5 7
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) |
9 |
|
simp23 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ∈ 𝐴 ) |
10 |
6 1 2
|
hlatlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) ) |
11 |
3 4 9 10
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) ) |
12 |
|
simp3r |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) |
13 |
11 12
|
breqtrrd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) |
14 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝐾 ∈ Lat ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
17 |
16 2
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
18 |
5 17
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
19 |
16 2
|
atbase |
⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
20 |
9 19
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
21 |
16 1 2
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
3 4 5 21
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
16 6 1
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) |
24 |
15 18 20 22 23
|
syl13anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( ( 𝑄 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) ) |
25 |
8 13 24
|
mpbi2and |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) |
26 |
|
simp3l |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → 𝑄 ≠ 𝑅 ) |
27 |
6 1 2
|
ps-1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ≠ 𝑅 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ↔ ( 𝑄 ∨ 𝑅 ) = ( 𝑃 ∨ 𝑄 ) ) ) |
28 |
3 5 9 26 4 5 27
|
syl132anc |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ↔ ( 𝑄 ∨ 𝑅 ) = ( 𝑃 ∨ 𝑄 ) ) ) |
29 |
25 28
|
mpbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑃 ∨ 𝑄 ) ) |
30 |
29
|
eqcomd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑃 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑅 ) ) |