Step |
Hyp |
Ref |
Expression |
1 |
|
lhpelim.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lhpelim.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
lhpelim.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
lhpelim.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
lhpelim.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
lhpelim.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
8 |
2 4 7 5 6
|
lhpmat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
9 |
8
|
3adant3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
10 |
9
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∧ 𝑊 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 0. ‘ 𝐾 ) ∨ ( 𝑋 ∧ 𝑊 ) ) ) |
11 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ HL ) |
12 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑃 ∈ 𝐴 ) |
13 |
11
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Lat ) |
14 |
|
simp3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
15 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑊 ∈ 𝐻 ) |
16 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
17 |
15 16
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑊 ∈ 𝐵 ) |
18 |
1 4
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
19 |
13 14 17 18
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) |
20 |
1 2 4
|
latmle2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
21 |
13 14 17 20
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) |
22 |
1 2 3 4 5
|
atmod4i2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) → ( ( 𝑃 ∧ 𝑊 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) ) |
23 |
11 12 19 17 21 22
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∧ 𝑊 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) ) |
24 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
25 |
11 24
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ OL ) |
26 |
1 3 7
|
olj02 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → ( ( 0. ‘ 𝐾 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑋 ∧ 𝑊 ) ) |
27 |
25 19 26
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 0. ‘ 𝐾 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑋 ∧ 𝑊 ) ) |
28 |
10 23 27
|
3eqtr3d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) ) |