Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem6.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem6.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem6.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihmeetlem6.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
5 |
|
dihmeetlem6.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dihmeetlem6.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
7 |
|
simprlr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ 𝑄 ≤ 𝑊 ) |
8 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ HL ) |
9 |
8
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝐾 ∈ Lat ) |
10 |
|
simpl2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑋 ∈ 𝐵 ) |
11 |
|
simpl3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑌 ∈ 𝐵 ) |
12 |
1 5
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
13 |
9 10 11 12
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
14 |
|
simprll |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐴 ) |
15 |
1 6
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 ) |
16 |
14 15
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ∈ 𝐵 ) |
17 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑊 ∈ 𝐻 ) |
18 |
1 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
19 |
17 18
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑊 ∈ 𝐵 ) |
20 |
1 2 4
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) ) |
21 |
9 13 16 19 20
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) ) |
22 |
|
simpr |
⊢ ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑄 ≤ 𝑊 ) → 𝑄 ≤ 𝑊 ) |
23 |
21 22
|
syl6bir |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 → 𝑄 ≤ 𝑊 ) ) |
24 |
7 23
|
mtod |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) |
25 |
|
simprr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → 𝑄 ≤ 𝑋 ) |
26 |
1 2 4 5 6
|
dihmeetlem5 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |
27 |
8 10 11 14 25 26
|
syl32anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ) |
28 |
27
|
breq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ( ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 ↔ ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑄 ) ≤ 𝑊 ) ) |
29 |
24 28
|
mtbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ 𝑋 ) ) → ¬ ( 𝑋 ∧ ( 𝑌 ∨ 𝑄 ) ) ≤ 𝑊 ) |