Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme11.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme11.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme11.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme11.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme11.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme11.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
cdleme11.c |
⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
8 |
|
cdleme11.d |
⊢ 𝐷 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) |
9 |
|
cdleme11.f |
⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
10 |
9
|
oveq2i |
⊢ ( 𝑄 ∨ 𝐹 ) = ( 𝑄 ∨ ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
11 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ HL ) |
12 |
|
simp22l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ 𝐴 ) |
13 |
11
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝐾 ∈ Lat ) |
14 |
|
simp23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑆 ∈ 𝐴 ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
16 |
15 4
|
atbase |
⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
17 |
14 16
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
18 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
19 |
|
simp21 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ 𝐴 ) |
20 |
1 2 3 4 5 6 15
|
cdleme0aa |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
21 |
18 19 12 20
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
22 |
15 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑆 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
13 17 21 22
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑆 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
15 4
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
25 |
12 24
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
26 |
15 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
27 |
19 26
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
28 |
15 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
29 |
13 27 17 28
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
30 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ 𝐻 ) |
31 |
15 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
32 |
30 31
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
33 |
15 3
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
34 |
13 29 32 33
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
35 |
15 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
36 |
13 25 34 35
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) |
37 |
15 1 2
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑄 ≤ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
38 |
13 25 34 37
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → 𝑄 ≤ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
39 |
15 1 2 3 4
|
atmod1i1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑄 ≤ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) → ( 𝑄 ∨ ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) = ( ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
40 |
11 12 23 36 38 39
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) = ( ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
41 |
10 40
|
syl5eq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ 𝐹 ) = ( ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
42 |
|
simp22 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
43 |
1 2 3 4 5 6
|
cdleme0cq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) ) |
44 |
18 19 42 43
|
syl12anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) ) |
45 |
44
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑆 ∨ ( 𝑄 ∨ 𝑈 ) ) = ( 𝑆 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
46 |
15 2
|
latj12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) = ( 𝑆 ∨ ( 𝑄 ∨ 𝑈 ) ) ) |
47 |
13 25 17 21 46
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) = ( 𝑆 ∨ ( 𝑄 ∨ 𝑈 ) ) ) |
48 |
15 2
|
latj13 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) = ( 𝑆 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
49 |
13 25 27 17 48
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) = ( 𝑆 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
50 |
45 47 49
|
3eqtr4d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ) |
51 |
50
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑄 ∨ ( 𝑆 ∨ 𝑈 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
52 |
15 1 3
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) ) |
53 |
13 29 32 52
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) ) |
54 |
15 1 2
|
latjlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ≤ ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ) ) |
55 |
13 34 29 25 54
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ≤ ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ) ) |
56 |
53 55
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ≤ ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ) |
57 |
15 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) |
58 |
13 25 29 57
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) |
59 |
15 1 3
|
latleeqm2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ≤ ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ↔ ( ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
60 |
13 36 58 59
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ≤ ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ↔ ( ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) |
61 |
56 60
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) |
62 |
7
|
oveq2i |
⊢ ( 𝑄 ∨ 𝐶 ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) |
63 |
61 62
|
eqtr4di |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( ( 𝑄 ∨ ( 𝑃 ∨ 𝑆 ) ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( 𝑄 ∨ 𝐶 ) ) |
64 |
41 51 63
|
3eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ 𝐹 ) = ( 𝑄 ∨ 𝐶 ) ) |