Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme17.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme17.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme17.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme17.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme17.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme17.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
cdleme17.f |
⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
8 |
|
cdleme17.g |
⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
9 |
|
cdleme17.c |
⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
10 |
|
simp33 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
12 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ HL ) |
13 |
12
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ Lat ) |
14 |
|
simpl32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑆 ∈ 𝐴 ) |
15 |
11 4
|
atbase |
⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
16 |
14 15
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
17 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
18 |
11 2 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
12 17 14 18
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
|
simpl31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
21 |
11 2 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
12 17 20 21
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
1 2 4
|
hlatlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑆 ≤ ( 𝑃 ∨ 𝑆 ) ) |
24 |
12 17 14 23
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑆 ≤ ( 𝑃 ∨ 𝑆 ) ) |
25 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑊 ∈ 𝐻 ) |
26 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑃 ≤ 𝑊 ) |
27 |
1 2 3 4 5 9
|
cdleme8 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝐶 ) = ( 𝑃 ∨ 𝑆 ) ) |
28 |
12 25 17 26 14 27
|
syl221anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝐶 ) = ( 𝑃 ∨ 𝑆 ) ) |
29 |
1 2 4
|
hlatlej1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
30 |
12 17 20 29
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
31 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) |
32 |
11 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
33 |
17 32
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
34 |
11 2 3 4 5 9
|
cdleme9b |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
35 |
12 17 14 25 34
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
36 |
11 1 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑃 ∨ 𝐶 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
37 |
13 33 35 22 36
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑃 ∨ 𝐶 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
38 |
30 31 37
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝐶 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
39 |
28 38
|
eqbrtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝑆 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
40 |
11 1 13 16 19 22 24 39
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
41 |
10 40
|
mtand |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) |