Step |
Hyp |
Ref |
Expression |
1 |
|
dalem.ph |
⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) ) |
2 |
|
dalem.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalem.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalem.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem.ps |
⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
6 |
|
dalem23.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
7 |
|
dalem23.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
8 |
|
dalem23.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
9 |
|
dalem23.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
10 |
|
dalem23.g |
⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) |
11 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
12 |
11
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ Lat ) |
13 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
14 |
13
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL ) |
15 |
5
|
dalemccea |
⊢ ( 𝜓 → 𝑐 ∈ 𝐴 ) |
16 |
15
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 ) |
17 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
18 |
17
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ∈ 𝐴 ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
20 |
19 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
14 16 18 20
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
5
|
dalemddea |
⊢ ( 𝜓 → 𝑑 ∈ 𝐴 ) |
23 |
22
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 ) |
24 |
1
|
dalemsea |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
25 |
24
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ∈ 𝐴 ) |
26 |
19 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
14 23 25 26
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
28 |
19 2 6
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ≤ ( 𝑐 ∨ 𝑃 ) ) |
29 |
12 21 27 28
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ≤ ( 𝑐 ∨ 𝑃 ) ) |
30 |
10 29
|
eqbrtrid |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≤ ( 𝑐 ∨ 𝑃 ) ) |
31 |
1 2 3 4 5 6 7 8 9 10
|
dalem23 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 ) |
32 |
1 2 3 4 7 8
|
dalemply |
⊢ ( 𝜑 → 𝑃 ≤ 𝑌 ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ≤ 𝑌 ) |
34 |
1 2 3 4 5 6 7 8 9 10
|
dalem24 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝐺 ≤ 𝑌 ) |
35 |
|
nbrne2 |
⊢ ( ( 𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌 ) → 𝑃 ≠ 𝐺 ) |
36 |
35
|
necomd |
⊢ ( ( 𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌 ) → 𝐺 ≠ 𝑃 ) |
37 |
33 34 36
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≠ 𝑃 ) |
38 |
2 3 4
|
hlatexch2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝐺 ≠ 𝑃 ) → ( 𝐺 ≤ ( 𝑐 ∨ 𝑃 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ) ) |
39 |
14 31 16 18 37 38
|
syl131anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ≤ ( 𝑐 ∨ 𝑃 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ) ) |
40 |
30 39
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ) |