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 |
1
|
4atexlemnslpq |
⊢ ( 𝜑 → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
10 |
1
|
4atexlemk |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
11 |
1
|
4atexlemp |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
12 |
1
|
4atexlems |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
13 |
2 3 5
|
hlatlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑆 ≤ ( 𝑃 ∨ 𝑆 ) ) |
14 |
10 11 12 13
|
syl3anc |
⊢ ( 𝜑 → 𝑆 ≤ ( 𝑃 ∨ 𝑆 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → 𝑆 ≤ ( 𝑃 ∨ 𝑆 ) ) |
16 |
1
|
4atexlemkl |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
17 |
1 3 5
|
4atexlempsb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
1 6
|
4atexlemwb |
⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
20 |
19 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) ) |
21 |
16 17 18 20
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑆 ) ) |
22 |
8 21
|
eqbrtrid |
⊢ ( 𝜑 → 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ) |
23 |
1
|
4atexlemkc |
⊢ ( 𝜑 → 𝐾 ∈ CvLat ) |
24 |
1 2 3 4 5 6 7 8
|
4atexlemv |
⊢ ( 𝜑 → 𝑉 ∈ 𝐴 ) |
25 |
19 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ 𝑊 ) |
26 |
16 17 18 25
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ≤ 𝑊 ) |
27 |
8 26
|
eqbrtrid |
⊢ ( 𝜑 → 𝑉 ≤ 𝑊 ) |
28 |
1
|
4atexlempw |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
29 |
28
|
simprd |
⊢ ( 𝜑 → ¬ 𝑃 ≤ 𝑊 ) |
30 |
|
nbrne2 |
⊢ ( ( 𝑉 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊 ) → 𝑉 ≠ 𝑃 ) |
31 |
27 29 30
|
syl2anc |
⊢ ( 𝜑 → 𝑉 ≠ 𝑃 ) |
32 |
2 3 5
|
cvlatexchb1 |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑉 ≠ 𝑃 ) → ( 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ↔ ( 𝑃 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑆 ) ) ) |
33 |
23 24 12 11 31 32
|
syl131anc |
⊢ ( 𝜑 → ( 𝑉 ≤ ( 𝑃 ∨ 𝑆 ) ↔ ( 𝑃 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑆 ) ) ) |
34 |
22 33
|
mpbid |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑆 ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝑃 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑆 ) ) |
36 |
|
oveq2 |
⊢ ( 𝑈 = 𝑉 → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑉 ) ) |
37 |
36
|
eqcomd |
⊢ ( 𝑈 = 𝑉 → ( 𝑃 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑈 ) ) |
38 |
1
|
4atexlemq |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
39 |
19 3 5
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
40 |
10 11 38 39
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
41 |
19 2 4
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
42 |
16 40 18 41
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
43 |
7 42
|
eqbrtrid |
⊢ ( 𝜑 → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ) |
44 |
1 2 3 4 5 6 7
|
4atexlemu |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
45 |
19 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ) |
46 |
16 40 18 45
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ) |
47 |
7 46
|
eqbrtrid |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |
48 |
|
nbrne2 |
⊢ ( ( 𝑈 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊 ) → 𝑈 ≠ 𝑃 ) |
49 |
47 29 48
|
syl2anc |
⊢ ( 𝜑 → 𝑈 ≠ 𝑃 ) |
50 |
2 3 5
|
cvlatexchb1 |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑈 ≠ 𝑃 ) → ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) ) ) |
51 |
23 44 38 11 49 50
|
syl131anc |
⊢ ( 𝜑 → ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) ) ) |
52 |
43 51
|
mpbid |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) ) |
53 |
37 52
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝑃 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) |
54 |
35 53
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → ( 𝑃 ∨ 𝑆 ) = ( 𝑃 ∨ 𝑄 ) ) |
55 |
15 54
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑈 = 𝑉 ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
56 |
55
|
ex |
⊢ ( 𝜑 → ( 𝑈 = 𝑉 → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
57 |
56
|
necon3bd |
⊢ ( 𝜑 → ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) → 𝑈 ≠ 𝑉 ) ) |
58 |
9 57
|
mpd |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |