Step |
Hyp |
Ref |
Expression |
1 |
|
trljat.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
trljat.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
trljat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
trljat.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
5 |
|
trljat.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
trljat.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL ) |
8 |
1 3 4 5
|
ltrnat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) |
9 |
8
|
3adant3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) |
10 |
7
|
hllatd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat ) |
11 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
13 |
12 3
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
14 |
11 13
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
15 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
16 |
|
simp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 ) |
17 |
12 4 5
|
ltrncl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
15 16 14 17
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
12 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
10 14 18 19
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 ) |
22 |
12 4
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
23 |
21 22
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
24 |
12 1 2
|
latlej2 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
25 |
10 14 18 24
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
26 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
27 |
12 1 2 26 3
|
atmod2i1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) |
28 |
7 9 20 23 25 27
|
syl131anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) |
29 |
1 3 4 5
|
ltrnel |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) |
30 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
31 |
1 2 30 3 4
|
lhpjat1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 1. ‘ 𝐾 ) ) |
32 |
7 21 29 31
|
syl21anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 1. ‘ 𝐾 ) ) |
33 |
32
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ∨ ( 𝐹 ‘ 𝑃 ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) ) |
34 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
35 |
7 34
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ OL ) |
36 |
12 26 30
|
olm11 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
37 |
35 20 36
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
38 |
28 33 37
|
3eqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
39 |
1 2 26 3 4 5 6
|
trlval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) |
40 |
39
|
oveq1d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ) |
41 |
12 4 5 6
|
trlcl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) |
42 |
15 16 41
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) |
43 |
12 2
|
latjcom |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ) |
44 |
10 42 18 43
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ) |
45 |
38 40 44
|
3eqtr2rd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) |