Step |
Hyp |
Ref |
Expression |
1 |
|
lhpat.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
lhpat.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
lhpat.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
lhpat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
lhpat.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
lhpat2.r |
⊢ 𝑅 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
simpl3r |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
8 |
|
simpr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑆 ≤ 𝑊 ) |
9 |
|
simp1ll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL ) |
10 |
9
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ Lat ) |
11 |
|
simp2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑆 ∈ 𝐴 ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
13 |
12 4
|
atbase |
⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
14 |
11 13
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
15 |
|
simp1rl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 ) |
16 |
|
simp2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 ) |
17 |
12 2 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
9 15 16 17
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
|
simp1lr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑊 ∈ 𝐻 ) |
20 |
12 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
21 |
19 20
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
22 |
12 1 3
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ 𝑊 ) ↔ 𝑆 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) |
23 |
10 14 18 21 22
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ 𝑊 ) ↔ 𝑆 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) |
24 |
23
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → ( ( 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ 𝑊 ) ↔ 𝑆 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) |
25 |
7 8 24
|
mpbi2and |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑆 ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) |
26 |
25 6
|
breqtrrdi |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑆 ≤ 𝑅 ) |
27 |
9
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝐾 ∈ HL ) |
28 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
29 |
27 28
|
syl |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝐾 ∈ AtLat ) |
30 |
|
simpl2r |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑆 ∈ 𝐴 ) |
31 |
|
simpl1l |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
32 |
|
simpl1r |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
33 |
|
simpl2l |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑄 ∈ 𝐴 ) |
34 |
|
simpl3l |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑃 ≠ 𝑄 ) |
35 |
1 2 3 4 5 6
|
lhpat2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑅 ∈ 𝐴 ) |
36 |
31 32 33 34 35
|
syl112anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑅 ∈ 𝐴 ) |
37 |
1 4
|
atcmp |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑆 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑆 ≤ 𝑅 ↔ 𝑆 = 𝑅 ) ) |
38 |
29 30 36 37
|
syl3anc |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → ( 𝑆 ≤ 𝑅 ↔ 𝑆 = 𝑅 ) ) |
39 |
26 38
|
mpbid |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑆 ≤ 𝑊 ) → 𝑆 = 𝑅 ) |
40 |
39
|
ex |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑆 ≤ 𝑊 → 𝑆 = 𝑅 ) ) |
41 |
12 1 3
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ) |
42 |
10 18 21 41
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ) |
43 |
6 42
|
eqbrtrid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑅 ≤ 𝑊 ) |
44 |
|
breq1 |
⊢ ( 𝑆 = 𝑅 → ( 𝑆 ≤ 𝑊 ↔ 𝑅 ≤ 𝑊 ) ) |
45 |
43 44
|
syl5ibrcom |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑆 = 𝑅 → 𝑆 ≤ 𝑊 ) ) |
46 |
40 45
|
impbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑆 ≤ 𝑊 ↔ 𝑆 = 𝑅 ) ) |
47 |
46
|
necon3bbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ¬ 𝑆 ≤ 𝑊 ↔ 𝑆 ≠ 𝑅 ) ) |