Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme0.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme0.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme0.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme0.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme0.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme0.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
7 |
6
|
oveq2i |
⊢ ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) |
8 |
|
simpll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ HL ) |
9 |
|
simprll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 ) |
10 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
11 |
10
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ Lat ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
13 |
12 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
14 |
9 13
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
15 |
|
simprr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 ) |
16 |
12 4
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
17 |
15 16
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
18 |
12 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
11 14 17 18
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
12 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
21 |
20
|
ad2antlr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
22 |
1 2 4
|
hlatlej1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
23 |
8 9 15 22
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
24 |
12 1 2 3 4
|
atmod3i1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑃 ∨ 𝑊 ) ) ) |
25 |
8 9 19 21 23 24
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑃 ∨ 𝑊 ) ) ) |
26 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
27 |
1 2 26 4 5
|
lhpjat2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) ) |
28 |
27
|
adantrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) ) |
29 |
28
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑃 ∨ 𝑊 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 1. ‘ 𝐾 ) ) ) |
30 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
31 |
30
|
ad2antrr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ OL ) |
32 |
12 3 26
|
olm11 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ 𝑄 ) ) |
33 |
31 19 32
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ 𝑄 ) ) |
34 |
25 29 33
|
3eqtrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( 𝑃 ∨ 𝑄 ) ) |
35 |
7 34
|
syl5eq |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) ) |