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 |
|
dalem-cly.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
6 |
|
dalem-cly.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
7 |
|
dalem-cly.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
8 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
9 |
1 4
|
dalemceb |
⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
10 |
1 5
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
12 |
11 2 3
|
latleeqj1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐶 ≤ 𝑌 ↔ ( 𝐶 ∨ 𝑌 ) = 𝑌 ) ) |
13 |
8 9 10 12
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ≤ 𝑌 ↔ ( 𝐶 ∨ 𝑌 ) = 𝑌 ) ) |
14 |
1
|
dalemclpjs |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ) |
15 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
16 |
1 2 3 4 5 6
|
dalemcea |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
17 |
1
|
dalemsea |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
18 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
19 |
1
|
dalemqea |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
20 |
1
|
dalem-clpjq |
⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) |
21 |
2 3 4
|
atnlej1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐶 ≠ 𝑃 ) |
22 |
15 16 18 19 20 21
|
syl131anc |
⊢ ( 𝜑 → 𝐶 ≠ 𝑃 ) |
23 |
2 3 4
|
hlatexch1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝐶 ≠ 𝑃 ) → ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) → 𝑆 ≤ ( 𝑃 ∨ 𝐶 ) ) ) |
24 |
15 16 17 18 22 23
|
syl131anc |
⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) → 𝑆 ≤ ( 𝑃 ∨ 𝐶 ) ) ) |
25 |
14 24
|
mpd |
⊢ ( 𝜑 → 𝑆 ≤ ( 𝑃 ∨ 𝐶 ) ) |
26 |
3 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐶 ∨ 𝑃 ) = ( 𝑃 ∨ 𝐶 ) ) |
27 |
15 16 18 26
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ 𝑃 ) = ( 𝑃 ∨ 𝐶 ) ) |
28 |
25 27
|
breqtrrd |
⊢ ( 𝜑 → 𝑆 ≤ ( 𝐶 ∨ 𝑃 ) ) |
29 |
1
|
dalemclqjt |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ) |
30 |
1
|
dalemtea |
⊢ ( 𝜑 → 𝑇 ∈ 𝐴 ) |
31 |
1
|
dalemrea |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
32 |
|
simp312 |
⊢ ( ( ( ( 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 ≤ ( 𝑄 ∨ 𝑅 ) ) |
33 |
1 32
|
sylbi |
⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ) |
34 |
2 3 4
|
atnlej1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ) → 𝐶 ≠ 𝑄 ) |
35 |
15 16 19 31 33 34
|
syl131anc |
⊢ ( 𝜑 → 𝐶 ≠ 𝑄 ) |
36 |
2 3 4
|
hlatexch1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝐶 ≠ 𝑄 ) → ( 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) → 𝑇 ≤ ( 𝑄 ∨ 𝐶 ) ) ) |
37 |
15 16 30 19 35 36
|
syl131anc |
⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) → 𝑇 ≤ ( 𝑄 ∨ 𝐶 ) ) ) |
38 |
29 37
|
mpd |
⊢ ( 𝜑 → 𝑇 ≤ ( 𝑄 ∨ 𝐶 ) ) |
39 |
3 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐶 ∨ 𝑄 ) = ( 𝑄 ∨ 𝐶 ) ) |
40 |
15 16 19 39
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ 𝑄 ) = ( 𝑄 ∨ 𝐶 ) ) |
41 |
38 40
|
breqtrrd |
⊢ ( 𝜑 → 𝑇 ≤ ( 𝐶 ∨ 𝑄 ) ) |
42 |
1 4
|
dalemseb |
⊢ ( 𝜑 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
43 |
11 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐶 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
44 |
15 16 18 43
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
45 |
1 4
|
dalemteb |
⊢ ( 𝜑 → 𝑇 ∈ ( Base ‘ 𝐾 ) ) |
46 |
11 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐶 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
47 |
15 16 19 46
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
48 |
11 2 3
|
latjlej12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐶 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑇 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐶 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑆 ≤ ( 𝐶 ∨ 𝑃 ) ∧ 𝑇 ≤ ( 𝐶 ∨ 𝑄 ) ) → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝐶 ∨ 𝑃 ) ∨ ( 𝐶 ∨ 𝑄 ) ) ) ) |
49 |
8 42 44 45 47 48
|
syl122anc |
⊢ ( 𝜑 → ( ( 𝑆 ≤ ( 𝐶 ∨ 𝑃 ) ∧ 𝑇 ≤ ( 𝐶 ∨ 𝑄 ) ) → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝐶 ∨ 𝑃 ) ∨ ( 𝐶 ∨ 𝑄 ) ) ) ) |
50 |
28 41 49
|
mp2and |
⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝐶 ∨ 𝑃 ) ∨ ( 𝐶 ∨ 𝑄 ) ) ) |
51 |
1 4
|
dalempeb |
⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
52 |
1 4
|
dalemqeb |
⊢ ( 𝜑 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
53 |
11 3
|
latjjdi |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐶 ∨ 𝑃 ) ∨ ( 𝐶 ∨ 𝑄 ) ) ) |
54 |
8 9 51 52 53
|
syl13anc |
⊢ ( 𝜑 → ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐶 ∨ 𝑃 ) ∨ ( 𝐶 ∨ 𝑄 ) ) ) |
55 |
50 54
|
breqtrrd |
⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ≤ ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ) |
56 |
1
|
dalemclrju |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) |
57 |
1
|
dalemuea |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
58 |
|
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 ) ) |
59 |
1 58
|
sylbi |
⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) |
60 |
2 3 4
|
atnlej1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) → 𝐶 ≠ 𝑅 ) |
61 |
15 16 31 18 59 60
|
syl131anc |
⊢ ( 𝜑 → 𝐶 ≠ 𝑅 ) |
62 |
2 3 4
|
hlatexch1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝐶 ≠ 𝑅 ) → ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) ) |
63 |
15 16 57 31 61 62
|
syl131anc |
⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) ) |
64 |
56 63
|
mpd |
⊢ ( 𝜑 → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) |
65 |
3 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝐶 ∨ 𝑅 ) = ( 𝑅 ∨ 𝐶 ) ) |
66 |
15 16 31 65
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ 𝑅 ) = ( 𝑅 ∨ 𝐶 ) ) |
67 |
64 66
|
breqtrrd |
⊢ ( 𝜑 → 𝑈 ≤ ( 𝐶 ∨ 𝑅 ) ) |
68 |
1 3 4
|
dalemsjteb |
⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) |
69 |
1 3 4
|
dalempjqeb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
70 |
11 3
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) |
71 |
8 9 69 70
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) |
72 |
1 4
|
dalemueb |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
73 |
11 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝐶 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) |
74 |
15 16 31 73
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) |
75 |
11 2 3
|
latjlej12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐶 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑆 ∨ 𝑇 ) ≤ ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝐶 ∨ 𝑅 ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ≤ ( ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∨ ( 𝐶 ∨ 𝑅 ) ) ) ) |
76 |
8 68 71 72 74 75
|
syl122anc |
⊢ ( 𝜑 → ( ( ( 𝑆 ∨ 𝑇 ) ≤ ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝐶 ∨ 𝑅 ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ≤ ( ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∨ ( 𝐶 ∨ 𝑅 ) ) ) ) |
77 |
55 67 76
|
mp2and |
⊢ ( 𝜑 → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ≤ ( ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∨ ( 𝐶 ∨ 𝑅 ) ) ) |
78 |
1 4
|
dalemreb |
⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
79 |
11 3
|
latjjdi |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝐶 ∨ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) = ( ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∨ ( 𝐶 ∨ 𝑅 ) ) ) |
80 |
8 9 69 78 79
|
syl13anc |
⊢ ( 𝜑 → ( 𝐶 ∨ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) = ( ( 𝐶 ∨ ( 𝑃 ∨ 𝑄 ) ) ∨ ( 𝐶 ∨ 𝑅 ) ) ) |
81 |
77 80
|
breqtrrd |
⊢ ( 𝜑 → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ≤ ( 𝐶 ∨ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) ) |
82 |
6
|
oveq2i |
⊢ ( 𝐶 ∨ 𝑌 ) = ( 𝐶 ∨ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
83 |
81 7 82
|
3brtr4g |
⊢ ( 𝜑 → 𝑍 ≤ ( 𝐶 ∨ 𝑌 ) ) |
84 |
|
breq2 |
⊢ ( ( 𝐶 ∨ 𝑌 ) = 𝑌 → ( 𝑍 ≤ ( 𝐶 ∨ 𝑌 ) ↔ 𝑍 ≤ 𝑌 ) ) |
85 |
83 84
|
syl5ibcom |
⊢ ( 𝜑 → ( ( 𝐶 ∨ 𝑌 ) = 𝑌 → 𝑍 ≤ 𝑌 ) ) |
86 |
13 85
|
sylbid |
⊢ ( 𝜑 → ( 𝐶 ≤ 𝑌 → 𝑍 ≤ 𝑌 ) ) |
87 |
1
|
dalemzeo |
⊢ ( 𝜑 → 𝑍 ∈ 𝑂 ) |
88 |
1
|
dalemyeo |
⊢ ( 𝜑 → 𝑌 ∈ 𝑂 ) |
89 |
2 5
|
lplncmp |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) → ( 𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌 ) ) |
90 |
15 87 88 89
|
syl3anc |
⊢ ( 𝜑 → ( 𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌 ) ) |
91 |
|
eqcom |
⊢ ( 𝑍 = 𝑌 ↔ 𝑌 = 𝑍 ) |
92 |
90 91
|
bitrdi |
⊢ ( 𝜑 → ( 𝑍 ≤ 𝑌 ↔ 𝑌 = 𝑍 ) ) |
93 |
86 92
|
sylibd |
⊢ ( 𝜑 → ( 𝐶 ≤ 𝑌 → 𝑌 = 𝑍 ) ) |
94 |
93
|
necon3ad |
⊢ ( 𝜑 → ( 𝑌 ≠ 𝑍 → ¬ 𝐶 ≤ 𝑌 ) ) |
95 |
94
|
imp |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ¬ 𝐶 ≤ 𝑌 ) |