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 |
|
4thatlem0.d |
⊢ 𝐷 = ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) |
11 |
1 2 3 4 5 6 7 8
|
4atexlemtlw |
⊢ ( 𝜑 → 𝑇 ≤ 𝑊 ) |
12 |
1 2 3 4 5 6 7 8 9
|
4atexlemnclw |
⊢ ( 𝜑 → ¬ 𝐶 ≤ 𝑊 ) |
13 |
|
nbrne2 |
⊢ ( ( 𝑇 ≤ 𝑊 ∧ ¬ 𝐶 ≤ 𝑊 ) → 𝑇 ≠ 𝐶 ) |
14 |
11 12 13
|
syl2anc |
⊢ ( 𝜑 → 𝑇 ≠ 𝐶 ) |
15 |
1
|
4atexlemk |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
16 |
1
|
4atexlemq |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
17 |
1
|
4atexlemt |
⊢ ( 𝜑 → 𝑇 ∈ 𝐴 ) |
18 |
3 5
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑇 ) = ( 𝑇 ∨ 𝑄 ) ) |
19 |
15 16 17 18
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) = ( 𝑇 ∨ 𝑄 ) ) |
20 |
|
simp221 |
⊢ ( ( ( ( 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 ) ) ) → 𝑅 ∈ 𝐴 ) |
21 |
1 20
|
sylbi |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
22 |
3 5
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → ( 𝑅 ∨ 𝑇 ) = ( 𝑇 ∨ 𝑅 ) ) |
23 |
15 21 17 22
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ∨ 𝑇 ) = ( 𝑇 ∨ 𝑅 ) ) |
24 |
19 23
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) = ( ( 𝑇 ∨ 𝑄 ) ∧ ( 𝑇 ∨ 𝑅 ) ) ) |
25 |
1
|
4atexlemkc |
⊢ ( 𝜑 → 𝐾 ∈ CvLat ) |
26 |
1
|
4atexlemp |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
27 |
1
|
4atexlempnq |
⊢ ( 𝜑 → 𝑃 ≠ 𝑄 ) |
28 |
|
simp223 |
⊢ ( ( ( ( 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 ) ) ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) |
29 |
1 28
|
sylbi |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) |
30 |
5 3
|
cvlsupr6 |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑅 ≠ 𝑄 ) |
31 |
30
|
necomd |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → 𝑄 ≠ 𝑅 ) |
32 |
25 26 16 21 27 29 31
|
syl132anc |
⊢ ( 𝜑 → 𝑄 ≠ 𝑅 ) |
33 |
1 2 3 4 5 6 7 8
|
4atexlemntlpq |
⊢ ( 𝜑 → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) |
34 |
5 3
|
cvlsupr7 |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) |
35 |
25 26 16 21 27 29 34
|
syl132anc |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) |
36 |
3 5
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) ) |
37 |
15 16 21 36
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑄 ) ) |
38 |
35 37
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑅 ) ) |
39 |
38
|
breq2d |
⊢ ( 𝜑 → ( 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑇 ≤ ( 𝑄 ∨ 𝑅 ) ) ) |
40 |
33 39
|
mtbid |
⊢ ( 𝜑 → ¬ 𝑇 ≤ ( 𝑄 ∨ 𝑅 ) ) |
41 |
2 3 4 5
|
2llnma2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ( ( 𝑇 ∨ 𝑄 ) ∧ ( 𝑇 ∨ 𝑅 ) ) = 𝑇 ) |
42 |
15 16 21 17 32 40 41
|
syl132anc |
⊢ ( 𝜑 → ( ( 𝑇 ∨ 𝑄 ) ∧ ( 𝑇 ∨ 𝑅 ) ) = 𝑇 ) |
43 |
24 42
|
eqtr2d |
⊢ ( 𝜑 → 𝑇 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) |
44 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝑇 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) |
45 |
1
|
4atexlemkl |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
46 |
1 3 5
|
4atexlemqtb |
⊢ ( 𝜑 → ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) |
47 |
1 3 5
|
4atexlempsb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
48 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
49 |
48 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) ≤ ( 𝑄 ∨ 𝑇 ) ) |
50 |
45 46 47 49
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) ≤ ( 𝑄 ∨ 𝑇 ) ) |
51 |
9 50
|
eqbrtrid |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ) |
52 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ) |
53 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐶 = 𝐷 ) |
54 |
48 3 5
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → ( 𝑅 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) |
55 |
15 21 17 54
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) |
56 |
48 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) ≤ ( 𝑅 ∨ 𝑇 ) ) |
57 |
45 55 47 56
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) ) ≤ ( 𝑅 ∨ 𝑇 ) ) |
58 |
10 57
|
eqbrtrid |
⊢ ( 𝜑 → 𝐷 ≤ ( 𝑅 ∨ 𝑇 ) ) |
59 |
58
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐷 ≤ ( 𝑅 ∨ 𝑇 ) ) |
60 |
53 59
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐶 ≤ ( 𝑅 ∨ 𝑇 ) ) |
61 |
1 2 3 4 5 6 7 8 9
|
4atexlemc |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
62 |
48 5
|
atbase |
⊢ ( 𝐶 ∈ 𝐴 → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
63 |
61 62
|
syl |
⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
64 |
48 2 4
|
latlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑇 ) ) ↔ 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) ) |
65 |
45 63 46 55 64
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑇 ) ) ↔ 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) ) |
66 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → ( ( 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑇 ) ) ↔ 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) ) |
67 |
52 60 66
|
mpbi2and |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) |
68 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
69 |
15 68
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ AtLat ) |
70 |
43 17
|
eqeltrrd |
⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ∈ 𝐴 ) |
71 |
2 5
|
atcmp |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝐶 ∈ 𝐴 ∧ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ∈ 𝐴 ) → ( 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ↔ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) ) |
72 |
69 61 70 71
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ↔ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → ( 𝐶 ≤ ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ↔ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) ) |
74 |
67 73
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑅 ∨ 𝑇 ) ) ) |
75 |
44 74
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐶 = 𝐷 ) → 𝑇 = 𝐶 ) |
76 |
75
|
ex |
⊢ ( 𝜑 → ( 𝐶 = 𝐷 → 𝑇 = 𝐶 ) ) |
77 |
76
|
necon3d |
⊢ ( 𝜑 → ( 𝑇 ≠ 𝐶 → 𝐶 ≠ 𝐷 ) ) |
78 |
14 77
|
mpd |
⊢ ( 𝜑 → 𝐶 ≠ 𝐷 ) |