Step |
Hyp |
Ref |
Expression |
1 |
|
dalema.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 |
|
dalemc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalemc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalemc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem5.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
6 |
|
dalem5.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
7 |
|
dalem5.w |
⊢ 𝑊 = ( 𝑌 ∨ 𝐶 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
9 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
10 |
1 4
|
dalemueb |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
11 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
12 |
1
|
dalemrea |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
13 |
1 2 3 4 5 6
|
dalemcea |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
14 |
8 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝑅 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) ) |
15 |
11 12 13 14
|
syl3anc |
⊢ ( 𝜑 → ( 𝑅 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) ) |
16 |
1 5
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
17 |
1 4
|
dalemceb |
⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
18 |
8 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑌 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
9 16 17 18
|
syl3anc |
⊢ ( 𝜑 → ( 𝑌 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
7 19
|
eqeltrid |
⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
21 |
1
|
dalemclrju |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) |
22 |
1
|
dalemuea |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
23 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
24 |
|
simp313 |
⊢ ( ( ( ( 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 ) ) ) ) → ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) |
25 |
1 24
|
sylbi |
⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) |
26 |
2 3 4
|
atnlej1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) → 𝐶 ≠ 𝑅 ) |
27 |
11 13 12 23 25 26
|
syl131anc |
⊢ ( 𝜑 → 𝐶 ≠ 𝑅 ) |
28 |
2 3 4
|
hlatexch1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝐶 ≠ 𝑅 ) → ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) ) |
29 |
11 13 22 12 27 28
|
syl131anc |
⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) ) |
30 |
21 29
|
mpd |
⊢ ( 𝜑 → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) |
31 |
1 3 4
|
dalempjqeb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
32 |
1 4
|
dalemreb |
⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
33 |
8 2 3
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → 𝑅 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
34 |
9 31 32 33
|
syl3anc |
⊢ ( 𝜑 → 𝑅 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
35 |
34 6
|
breqtrrdi |
⊢ ( 𝜑 → 𝑅 ≤ 𝑌 ) |
36 |
8 2 3
|
latjlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑅 ≤ 𝑌 → ( 𝑅 ∨ 𝐶 ) ≤ ( 𝑌 ∨ 𝐶 ) ) ) |
37 |
9 32 16 17 36
|
syl13anc |
⊢ ( 𝜑 → ( 𝑅 ≤ 𝑌 → ( 𝑅 ∨ 𝐶 ) ≤ ( 𝑌 ∨ 𝐶 ) ) ) |
38 |
35 37
|
mpd |
⊢ ( 𝜑 → ( 𝑅 ∨ 𝐶 ) ≤ ( 𝑌 ∨ 𝐶 ) ) |
39 |
38 7
|
breqtrrdi |
⊢ ( 𝜑 → ( 𝑅 ∨ 𝐶 ) ≤ 𝑊 ) |
40 |
8 2 9 10 15 20 30 39
|
lattrd |
⊢ ( 𝜑 → 𝑈 ≤ 𝑊 ) |