Step |
Hyp |
Ref |
Expression |
1 |
|
atbtwn.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
atbtwn.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
atbtwn.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
atbtwn.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
simpr2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≤ 𝑋 ) |
6 |
|
simpr3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ¬ 𝑄 ≤ 𝑋 ) |
7 |
|
nbrne2 |
⊢ ( ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) → 𝑃 ≠ 𝑄 ) |
8 |
5 6 7
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → 𝑃 ≠ 𝑄 ) |
9 |
2 3 4
|
hlsupr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
10 |
8 9
|
syldan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
11 |
|
simp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑟 ≠ 𝑄 ) |
12 |
|
simp31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑟 ≠ 𝑃 ) |
13 |
|
simp1l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) |
14 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑟 ∈ 𝐴 ) |
15 |
|
simp1r1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑋 ∈ 𝐵 ) |
16 |
|
simp1r2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≤ 𝑋 ) |
17 |
|
simp1r3 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑄 ≤ 𝑋 ) |
18 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) |
19 |
1 2 3 4
|
atbtwn |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑟 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑟 ≠ 𝑃 ↔ ¬ 𝑟 ≤ 𝑋 ) ) |
20 |
13 14 15 16 17 18 19
|
syl123anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑟 ≠ 𝑃 ↔ ¬ 𝑟 ≤ 𝑋 ) ) |
21 |
12 20
|
mpbid |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑟 ≤ 𝑋 ) |
22 |
11 21 18
|
3jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ∧ 𝑟 ∈ 𝐴 ∧ ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
23 |
22
|
3exp |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( 𝑟 ∈ 𝐴 → ( ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ) |
24 |
23
|
reximdvai |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ( ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑄 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) |
25 |
10 24
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) → ∃ 𝑟 ∈ 𝐴 ( 𝑟 ≠ 𝑄 ∧ ¬ 𝑟 ≤ 𝑋 ∧ 𝑟 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |