Step |
Hyp |
Ref |
Expression |
1 |
|
2atnelpln.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
2 |
|
2atnelpln.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
2atnelpln.p |
⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) |
4 |
|
hllat |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → 𝐾 ∈ Lat ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
6 1 2
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) |
8 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
9 |
6 8
|
latref |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) |
10 |
5 7 9
|
syl2anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) |
11 |
|
simpl1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) → 𝐾 ∈ HL ) |
12 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) → ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) |
13 |
|
simpl2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) → 𝑄 ∈ 𝐴 ) |
14 |
|
simpl3 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) → 𝑅 ∈ 𝐴 ) |
15 |
8 1 2 3
|
lplnnle2at |
⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ¬ ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) |
16 |
11 12 13 14 15
|
syl13anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) → ¬ ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) |
17 |
16
|
ex |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 → ¬ ( 𝑄 ∨ 𝑅 ) ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) ) |
18 |
10 17
|
mt2d |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ¬ ( 𝑄 ∨ 𝑅 ) ∈ 𝑃 ) |