Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem7.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem7.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem7.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihmeetlem7.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
dihmeetlem7.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
simprr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ¬ 𝑝 ≤ 𝑌 ) |
7 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ HL ) |
8 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
9 |
7 8
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ AtLat ) |
10 |
|
simprl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝑝 ∈ 𝐴 ) |
11 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝑌 ∈ 𝐵 ) |
12 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
13 |
1 2 4 12 5
|
atnle |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( ¬ 𝑝 ≤ 𝑌 ↔ ( 𝑝 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) ) |
14 |
9 10 11 13
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ¬ 𝑝 ≤ 𝑌 ↔ ( 𝑝 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) ) |
15 |
6 14
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( 𝑝 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) |
16 |
15
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑝 ∧ 𝑌 ) ) = ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) ) |
17 |
7
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ Lat ) |
18 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝑋 ∈ 𝐵 ) |
19 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
20 |
17 18 11 19
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) |
21 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) |
22 |
17 18 11 21
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) |
23 |
1 2 3 4 5
|
atmod1i2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑝 ∧ 𝑌 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) ) |
24 |
7 10 20 11 22 23
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 𝑝 ∧ 𝑌 ) ) = ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) ) |
25 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
26 |
7 25
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → 𝐾 ∈ OL ) |
27 |
1 3 12
|
olj01 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) ) |
28 |
26 20 27
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( 𝑋 ∧ 𝑌 ) ∨ ( 0. ‘ 𝐾 ) ) = ( 𝑋 ∧ 𝑌 ) ) |
29 |
16 24 28
|
3eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌 ) ) → ( ( ( 𝑋 ∧ 𝑌 ) ∨ 𝑝 ) ∧ 𝑌 ) = ( 𝑋 ∧ 𝑌 ) ) |