Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme0.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme0.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme0.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme0.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
8 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
9 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 ) |
10 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ≠ 𝑄 ) |
11 |
1 2 3 4 5 6
|
lhpat2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑈 ∈ 𝐴 ) |
12 |
7 8 9 10 11
|
syl112anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ∈ 𝐴 ) |
13 |
|
simp2ll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ 𝐴 ) |
14 |
1 2 3 4 5 6
|
cdlemeulpq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) |
15 |
7 13 9 14
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) |
16 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ HL ) |
17 |
16
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ Lat ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
19 |
18 2 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
16 13 9 19
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ 𝐻 ) |
22 |
18 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
23 |
21 22
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
24 |
18 1 3
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ) |
25 |
17 20 23 24
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ) |
26 |
6 25
|
eqbrtrid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ≤ 𝑊 ) |
27 |
|
breq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
28 |
|
breq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 ≤ 𝑊 ↔ 𝑈 ≤ 𝑊 ) ) |
29 |
27 28
|
anbi12d |
⊢ ( 𝑢 = 𝑈 → ( ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑢 ≤ 𝑊 ) ↔ ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ 𝑊 ) ) ) |
30 |
29
|
rspcev |
⊢ ( ( 𝑈 ∈ 𝐴 ∧ ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ 𝑊 ) ) → ∃ 𝑢 ∈ 𝐴 ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑢 ≤ 𝑊 ) ) |
31 |
12 15 26 30
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑢 ∈ 𝐴 ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑢 ≤ 𝑊 ) ) |