Step |
Hyp |
Ref |
Expression |
1 |
|
lhpat.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
lhpat.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
lhpat.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
lhpat.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
lhpat.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝐾 ∈ HL ) |
7 |
|
simp2l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ 𝐴 ) |
8 |
|
simp3l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ 𝐴 ) |
9 |
|
simp1r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑊 ∈ 𝐻 ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
11 |
10 5
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
12 |
9 11
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
13 |
|
simp3r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ≠ 𝑄 ) |
14 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
15 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
16 |
14 15 5
|
lhp1cvr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
17 |
16
|
3ad2ant1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
18 |
|
simp2r |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ¬ 𝑃 ≤ 𝑊 ) |
19 |
10 1 2 3 14 15 4
|
1cvrat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 ) |
20 |
6 7 8 12 13 17 18 19
|
syl133anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 ) |