Step |
Hyp |
Ref |
Expression |
1 |
|
4thatlem.ph |
⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) |
2 |
|
4thatlem0.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
4thatlem0.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
4thatlem0.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
4thatlem0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
4thatlem0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
4thatlem0.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
4thatlem0.v |
⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
9 |
|
4thatlem0.c |
⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) |
10 |
1
|
4atexlemkl |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
11 |
1 3 5
|
4atexlemqtb |
⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) |
12 |
1 3 5
|
4atexlempsb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
14 |
13 4
|
latmcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ) |
15 |
10 11 12 14
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ) |
16 |
9 15
|
syl5eq |
⊢ ( 𝜑 → 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ) |
17 |
1
|
4atexlemk |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
18 |
1
|
4atexlemp |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
19 |
1
|
4atexlems |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
20 |
1
|
4atexlemq |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
21 |
1
|
4atexlemt |
⊢ ( 𝜑 → 𝑇 ∈ 𝐴 ) |
22 |
1 2 3 5
|
4atexlempns |
⊢ ( 𝜑 → 𝑃 ≠ 𝑆 ) |
23 |
1 2 3 4 5 6 7 8
|
4atexlemntlpq |
⊢ ( 𝜑 → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) |
24 |
2 3 5
|
atnlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑇 ≠ 𝑄 ) |
25 |
24
|
necomd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ≠ 𝑇 ) |
26 |
17 21 18 20 23 25
|
syl131anc |
⊢ ( 𝜑 → 𝑄 ≠ 𝑇 ) |
27 |
1
|
4atexlempnq |
⊢ ( 𝜑 → 𝑃 ≠ 𝑄 ) |
28 |
1
|
4atexlemnslpq |
⊢ ( 𝜑 → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
29 |
2 3 5
|
4atlem0ae |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ) |
30 |
17 18 20 19 27 28 29
|
syl132anc |
⊢ ( 𝜑 → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ) |
31 |
13 5
|
atbase |
⊢ ( 𝑇 ∈ 𝐴 → 𝑇 ∈ ( Base ‘ 𝐾 ) ) |
32 |
21 31
|
syl |
⊢ ( 𝜑 → 𝑇 ∈ ( Base ‘ 𝐾 ) ) |
33 |
1 2 3 4 5 6 7
|
4atexlemu |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
34 |
1 2 3 4 5 6 7 8
|
4atexlemv |
⊢ ( 𝜑 → 𝑉 ∈ 𝐴 ) |
35 |
13 3 5
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) |
36 |
17 33 34 35
|
syl3anc |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) |
37 |
13 5
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
38 |
20 37
|
syl |
⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
39 |
13 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
40 |
10 12 38 39
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
41 |
1
|
4atexlemkc |
⊢ ( 𝜑 → 𝐾 ∈ CvLat ) |
42 |
1 2 3 4 5 6 7 8
|
4atexlemunv |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
43 |
1
|
4atexlemutvt |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) |
44 |
5 2 3
|
cvlsupr4 |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ ( 𝑈 ≠ 𝑉 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) → 𝑇 ≤ ( 𝑈 ∨ 𝑉 ) ) |
45 |
41 33 34 21 42 43 44
|
syl132anc |
⊢ ( 𝜑 → 𝑇 ≤ ( 𝑈 ∨ 𝑉 ) ) |
46 |
13 3 5
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
47 |
17 18 20 46
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
48 |
1 6
|
4atexlemwb |
⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
49 |
13 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
50 |
10 47 48 49
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
51 |
7 50
|
eqbrtrid |
⊢ ( 𝜑 → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) |
52 |
13 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) ) |
53 |
10 12 48 52
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) ) |
54 |
8 53
|
eqbrtrid |
⊢ ( 𝜑 → 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ) |
55 |
13 5
|
atbase |
⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
56 |
33 55
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
57 |
13 5
|
atbase |
⊢ ( 𝑉 ∈ 𝐴 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
58 |
34 57
|
syl |
⊢ ( 𝜑 → 𝑉 ∈ ( Base ‘ 𝐾 ) ) |
59 |
13 2 3
|
latjlej12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ) → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) ) |
60 |
10 56 47 58 12 59
|
syl122anc |
⊢ ( 𝜑 → ( ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ) → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) ) |
61 |
51 54 60
|
mp2and |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) |
62 |
3 5
|
hlatjass |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) ) |
63 |
17 18 20 19 62
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) ) |
64 |
13 5
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
65 |
18 64
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
66 |
13 5
|
atbase |
⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
67 |
19 66
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
68 |
13 3
|
latj32 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) |
69 |
10 65 38 67 68
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑆 ) = ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) |
70 |
13 3
|
latjjdi |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) |
71 |
10 65 38 67 70
|
syl13anc |
⊢ ( 𝜑 → ( 𝑃 ∨ ( 𝑄 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) ) |
72 |
63 69 71
|
3eqtr3rd |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) |
73 |
61 72
|
breqtrd |
⊢ ( 𝜑 → ( 𝑈 ∨ 𝑉 ) ≤ ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) |
74 |
13 2 10 32 36 40 45 73
|
lattrd |
⊢ ( 𝜑 → 𝑇 ≤ ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) |
75 |
2 3 4 5
|
2atmat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ∧ ( 𝑄 ≠ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝑇 ≤ ( ( 𝑃 ∨ 𝑆 ) ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 ) |
76 |
17 18 19 20 21 22 26 30 74 75
|
syl333anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝑇 ) ) ∈ 𝐴 ) |
77 |
16 76
|
eqeltrd |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |