Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme11.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme11.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme11.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme11.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme11.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme11.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
|
cdleme11.c |
⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
8 |
|
cdleme11.d |
⊢ 𝐷 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) |
9 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
11 |
|
simp22 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑇 ∈ 𝐴 ) |
12 |
|
simp21 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) |
13 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐾 ∈ HL ) |
14 |
13
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐾 ∈ Lat ) |
15 |
|
simp12l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ∈ 𝐴 ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
17 |
16 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
18 |
15 17
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
19 |
|
simp21l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑆 ∈ 𝐴 ) |
20 |
16 4
|
atbase |
⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
21 |
19 20
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) ) |
22 |
16 4
|
atbase |
⊢ ( 𝑇 ∈ 𝐴 → 𝑇 ∈ ( Base ‘ 𝐾 ) ) |
23 |
11 22
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑇 ∈ ( Base ‘ 𝐾 ) ) |
24 |
|
simp1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) |
25 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) |
26 |
|
simp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) |
27 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) |
28 |
1 2 3 4 5 6
|
cdleme11c |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑃 ≤ ( 𝑆 ∨ 𝑇 ) ) |
29 |
24 25 26 27 28
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑃 ≤ ( 𝑆 ∨ 𝑇 ) ) |
30 |
16 1 2
|
latnlej1r |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑇 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑃 ≤ ( 𝑆 ∨ 𝑇 ) ) → 𝑃 ≠ 𝑇 ) |
31 |
14 18 21 23 29 30
|
syl131anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ≠ 𝑇 ) |
32 |
|
simp31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑆 ≠ 𝑇 ) |
33 |
1 2 4
|
hlatcon2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑃 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑇 ) ) |
34 |
13 19 11 15 32 29 33
|
syl132anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑇 ) ) |
35 |
1 2 3 4 5 8 7
|
cdleme0e |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑇 ) ) ) → 𝐷 ≠ 𝐶 ) |
36 |
9 10 11 12 31 34 35
|
syl132anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐷 ≠ 𝐶 ) |
37 |
36
|
necomd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐶 ≠ 𝐷 ) |