Step |
Hyp |
Ref |
Expression |
1 |
|
lplnexat.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
lplnexat.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
lplnexat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
lplnexat.n |
⊢ 𝑁 = ( LLines ‘ 𝐾 ) |
5 |
|
lplnexat.p |
⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) |
6 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → 𝑋 ∈ 𝑃 ) |
7 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → 𝐾 ∈ HL ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
9 |
8 5
|
lplnbase |
⊢ ( 𝑋 ∈ 𝑃 → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
10 |
6 9
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
11 |
8 1 2 3 4 5
|
islpln3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑧 ∈ 𝑁 ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) |
12 |
7 10 11
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑧 ∈ 𝑁 ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) |
13 |
6 12
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ∃ 𝑧 ∈ 𝑁 ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) |
14 |
|
simpll1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝐾 ∈ HL ) |
15 |
|
simpr2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑧 ∈ 𝑁 ) |
16 |
|
simpll3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑄 ∈ 𝐴 ) |
17 |
|
simpr1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑄 ≤ 𝑧 ) |
18 |
1 2 3 4
|
llnexatN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑧 ∈ 𝑁 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑧 ) → ∃ 𝑠 ∈ 𝐴 ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) |
19 |
14 15 16 17 18
|
syl31anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ∃ 𝑠 ∈ 𝐴 ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) |
20 |
|
simp1l1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝐾 ∈ HL ) |
21 |
|
simp22r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑟 ∈ 𝐴 ) |
22 |
|
simp3l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑠 ∈ 𝐴 ) |
23 |
|
simp1l3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑄 ∈ 𝐴 ) |
24 |
|
simp23l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ¬ 𝑟 ≤ 𝑧 ) |
25 |
|
simp3rr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑧 = ( 𝑄 ∨ 𝑠 ) ) |
26 |
25
|
breq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ( 𝑟 ≤ 𝑧 ↔ 𝑟 ≤ ( 𝑄 ∨ 𝑠 ) ) ) |
27 |
24 26
|
mtbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ¬ 𝑟 ≤ ( 𝑄 ∨ 𝑠 ) ) |
28 |
1 2 3
|
atnlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ) ∧ ¬ 𝑟 ≤ ( 𝑄 ∨ 𝑠 ) ) → 𝑟 ≠ 𝑠 ) |
29 |
20 21 23 22 27 28
|
syl131anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑟 ≠ 𝑠 ) |
30 |
2 3 4
|
llni2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ) ∧ 𝑟 ≠ 𝑠 ) → ( 𝑟 ∨ 𝑠 ) ∈ 𝑁 ) |
31 |
20 21 22 29 30
|
syl31anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ( 𝑟 ∨ 𝑠 ) ∈ 𝑁 ) |
32 |
|
simp3rl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑄 ≠ 𝑠 ) |
33 |
1 2 3
|
hlatcon2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑠 ∧ ¬ 𝑟 ≤ ( 𝑄 ∨ 𝑠 ) ) ) → ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑠 ) ) |
34 |
20 23 22 21 32 27 33
|
syl132anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑠 ) ) |
35 |
|
simp23r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑋 = ( 𝑧 ∨ 𝑟 ) ) |
36 |
25
|
oveq1d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ( 𝑧 ∨ 𝑟 ) = ( ( 𝑄 ∨ 𝑠 ) ∨ 𝑟 ) ) |
37 |
20
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝐾 ∈ Lat ) |
38 |
8 3
|
atbase |
⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
39 |
23 38
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
40 |
8 3
|
atbase |
⊢ ( 𝑠 ∈ 𝐴 → 𝑠 ∈ ( Base ‘ 𝐾 ) ) |
41 |
22 40
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑠 ∈ ( Base ‘ 𝐾 ) ) |
42 |
8 3
|
atbase |
⊢ ( 𝑟 ∈ 𝐴 → 𝑟 ∈ ( Base ‘ 𝐾 ) ) |
43 |
21 42
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑟 ∈ ( Base ‘ 𝐾 ) ) |
44 |
8 2
|
latj31 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑠 ∈ ( Base ‘ 𝐾 ) ∧ 𝑟 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ∨ 𝑠 ) ∨ 𝑟 ) = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) |
45 |
37 39 41 43 44
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ( ( 𝑄 ∨ 𝑠 ) ∨ 𝑟 ) = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) |
46 |
35 36 45
|
3eqtrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → 𝑋 = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) |
47 |
|
breq2 |
⊢ ( 𝑦 = ( 𝑟 ∨ 𝑠 ) → ( 𝑄 ≤ 𝑦 ↔ 𝑄 ≤ ( 𝑟 ∨ 𝑠 ) ) ) |
48 |
47
|
notbid |
⊢ ( 𝑦 = ( 𝑟 ∨ 𝑠 ) → ( ¬ 𝑄 ≤ 𝑦 ↔ ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑠 ) ) ) |
49 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑟 ∨ 𝑠 ) → ( 𝑦 ∨ 𝑄 ) = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) |
50 |
49
|
eqeq2d |
⊢ ( 𝑦 = ( 𝑟 ∨ 𝑠 ) → ( 𝑋 = ( 𝑦 ∨ 𝑄 ) ↔ 𝑋 = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) ) |
51 |
48 50
|
anbi12d |
⊢ ( 𝑦 = ( 𝑟 ∨ 𝑠 ) → ( ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ↔ ( ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑠 ) ∧ 𝑋 = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) ) ) |
52 |
51
|
rspcev |
⊢ ( ( ( 𝑟 ∨ 𝑠 ) ∈ 𝑁 ∧ ( ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑠 ) ∧ 𝑋 = ( ( 𝑟 ∨ 𝑠 ) ∨ 𝑄 ) ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) |
53 |
31 34 46 52
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) |
54 |
53
|
3expia |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( ( 𝑠 ∈ 𝐴 ∧ ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) |
55 |
54
|
expd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( 𝑠 ∈ 𝐴 → ( ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) ) |
56 |
55
|
rexlimdv |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( ∃ 𝑠 ∈ 𝐴 ( 𝑄 ≠ 𝑠 ∧ 𝑧 = ( 𝑄 ∨ 𝑠 ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) |
57 |
19 56
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) |
58 |
57
|
3exp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ( 𝑄 ≤ 𝑧 → ( ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) → ( ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) ) ) |
59 |
|
simpr2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑧 ∈ 𝑁 ) |
60 |
|
simpr1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ¬ 𝑄 ≤ 𝑧 ) |
61 |
|
simpll1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝐾 ∈ HL ) |
62 |
61
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝐾 ∈ Lat ) |
63 |
8 4
|
llnbase |
⊢ ( 𝑧 ∈ 𝑁 → 𝑧 ∈ ( Base ‘ 𝐾 ) ) |
64 |
59 63
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑧 ∈ ( Base ‘ 𝐾 ) ) |
65 |
|
simpr2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑟 ∈ 𝐴 ) |
66 |
65 42
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑟 ∈ ( Base ‘ 𝐾 ) ) |
67 |
8 1 2
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ∧ 𝑟 ∈ ( Base ‘ 𝐾 ) ) → 𝑧 ≤ ( 𝑧 ∨ 𝑟 ) ) |
68 |
62 64 66 67
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑧 ≤ ( 𝑧 ∨ 𝑟 ) ) |
69 |
|
simpr3r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑋 = ( 𝑧 ∨ 𝑟 ) ) |
70 |
68 69
|
breqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑧 ≤ 𝑋 ) |
71 |
|
simplr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑄 ≤ 𝑋 ) |
72 |
|
simpll3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑄 ∈ 𝐴 ) |
73 |
72 38
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) ) |
74 |
|
simpll2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑋 ∈ 𝑃 ) |
75 |
74 9
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑋 ∈ ( Base ‘ 𝐾 ) ) |
76 |
8 1 2
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑧 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑧 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋 ) ↔ ( 𝑧 ∨ 𝑄 ) ≤ 𝑋 ) ) |
77 |
62 64 73 75 76
|
syl13anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( ( 𝑧 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋 ) ↔ ( 𝑧 ∨ 𝑄 ) ≤ 𝑋 ) ) |
78 |
70 71 77
|
mpbi2and |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( 𝑧 ∨ 𝑄 ) ≤ 𝑋 ) |
79 |
8 2
|
latjcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑧 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
80 |
62 64 73 79
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( 𝑧 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
81 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
82 |
8 1 2 81 3
|
cvr1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑧 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ) → ( ¬ 𝑄 ≤ 𝑧 ↔ 𝑧 ( ⋖ ‘ 𝐾 ) ( 𝑧 ∨ 𝑄 ) ) ) |
83 |
61 64 72 82
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( ¬ 𝑄 ≤ 𝑧 ↔ 𝑧 ( ⋖ ‘ 𝐾 ) ( 𝑧 ∨ 𝑄 ) ) ) |
84 |
60 83
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑧 ( ⋖ ‘ 𝐾 ) ( 𝑧 ∨ 𝑄 ) ) |
85 |
8 81 4 5
|
lplni |
⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑧 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑧 ∈ 𝑁 ) ∧ 𝑧 ( ⋖ ‘ 𝐾 ) ( 𝑧 ∨ 𝑄 ) ) → ( 𝑧 ∨ 𝑄 ) ∈ 𝑃 ) |
86 |
61 80 59 84 85
|
syl31anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( 𝑧 ∨ 𝑄 ) ∈ 𝑃 ) |
87 |
1 5
|
lplncmp |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑧 ∨ 𝑄 ) ∈ 𝑃 ∧ 𝑋 ∈ 𝑃 ) → ( ( 𝑧 ∨ 𝑄 ) ≤ 𝑋 ↔ ( 𝑧 ∨ 𝑄 ) = 𝑋 ) ) |
88 |
61 86 74 87
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( ( 𝑧 ∨ 𝑄 ) ≤ 𝑋 ↔ ( 𝑧 ∨ 𝑄 ) = 𝑋 ) ) |
89 |
78 88
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ( 𝑧 ∨ 𝑄 ) = 𝑋 ) |
90 |
89
|
eqcomd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → 𝑋 = ( 𝑧 ∨ 𝑄 ) ) |
91 |
|
breq2 |
⊢ ( 𝑦 = 𝑧 → ( 𝑄 ≤ 𝑦 ↔ 𝑄 ≤ 𝑧 ) ) |
92 |
91
|
notbid |
⊢ ( 𝑦 = 𝑧 → ( ¬ 𝑄 ≤ 𝑦 ↔ ¬ 𝑄 ≤ 𝑧 ) ) |
93 |
|
oveq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∨ 𝑄 ) = ( 𝑧 ∨ 𝑄 ) ) |
94 |
93
|
eqeq2d |
⊢ ( 𝑦 = 𝑧 → ( 𝑋 = ( 𝑦 ∨ 𝑄 ) ↔ 𝑋 = ( 𝑧 ∨ 𝑄 ) ) ) |
95 |
92 94
|
anbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ↔ ( ¬ 𝑄 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑄 ) ) ) ) |
96 |
95
|
rspcev |
⊢ ( ( 𝑧 ∈ 𝑁 ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑄 ) ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) |
97 |
59 60 90 96
|
syl12anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) ∧ ( ¬ 𝑄 ≤ 𝑧 ∧ ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) ∧ ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) |
98 |
97
|
3exp2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ( ¬ 𝑄 ≤ 𝑧 → ( ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) → ( ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) ) ) |
99 |
58 98
|
pm2.61d |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ( ( 𝑧 ∈ 𝑁 ∧ 𝑟 ∈ 𝐴 ) → ( ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) ) |
100 |
99
|
rexlimdvv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ( ∃ 𝑧 ∈ 𝑁 ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑧 ∧ 𝑋 = ( 𝑧 ∨ 𝑟 ) ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) ) |
101 |
13 100
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑄 ≤ 𝑋 ) → ∃ 𝑦 ∈ 𝑁 ( ¬ 𝑄 ≤ 𝑦 ∧ 𝑋 = ( 𝑦 ∨ 𝑄 ) ) ) |