Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg8.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdlemg8.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdlemg8.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdlemg8.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdlemg8.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdlemg8.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
cdlemg10.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
simp33 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
9 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
10 |
|
simpl31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 ) |
11 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
12 |
1 2 3 4 5 6 7
|
trlval2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) |
13 |
9 10 11 12
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
15 |
|
simpl1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝐾 ∈ HL ) |
16 |
15
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝐾 ∈ Lat ) |
17 |
|
simp2ll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 ) |
18 |
17
|
adantr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 ) |
19 |
14 4
|
atbase |
⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
20 |
18 19
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
21 |
14 5 6
|
ltrncl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
22 |
9 10 20 21
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
14 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
16 20 22 23
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) |
25 |
|
simpl1r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑊 ∈ 𝐻 ) |
26 |
14 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
27 |
25 26
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
28 |
14 3
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
29 |
16 24 27 28
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) |
30 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 ) |
31 |
14 4
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
32 |
30 31
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
33 |
14 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
34 |
16 20 32 33
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
35 |
14 1 3
|
latmle1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) |
36 |
16 24 27 35
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ) |
37 |
14 1 2
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
38 |
16 20 32 37
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) |
39 |
14 5 6
|
ltrncl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
40 |
9 10 32 39
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
41 |
14 1 2
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺 ‘ 𝑃 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) |
42 |
16 22 40 41
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) |
43 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) |
44 |
42 43
|
breqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
45 |
14 1 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
46 |
16 20 22 34 45
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
47 |
38 44 46
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
48 |
14 1 16 29 24 34 36 47
|
lattrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
49 |
13 48
|
eqbrtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
50 |
49
|
ex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
51 |
50
|
necon3bd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ≠ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ) |
52 |
8 51
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ 𝑄 ) ≠ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) |