Step |
Hyp |
Ref |
Expression |
1 |
|
lhp2a.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
lhp2a.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
lhp2a.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
5 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
6 |
4 5 3
|
lhp1cvr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
7 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ HL ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
9 |
8 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
11 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
12 |
8 4
|
op1cl |
⊢ ( 𝐾 ∈ OP → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
11 12
|
syl |
⊢ ( 𝐾 ∈ HL → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) |
15 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
16 |
8 1 15 5 2
|
cvrval3 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ ( 1. ‘ 𝐾 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑊 ( join ‘ 𝐾 ) 𝑝 ) = ( 1. ‘ 𝐾 ) ) ) ) |
17 |
7 10 14 16
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑊 ( join ‘ 𝐾 ) 𝑝 ) = ( 1. ‘ 𝐾 ) ) ) ) |
18 |
6 17
|
mpbid |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑊 ( join ‘ 𝐾 ) 𝑝 ) = ( 1. ‘ 𝐾 ) ) ) |
19 |
|
simpl |
⊢ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑊 ( join ‘ 𝐾 ) 𝑝 ) = ( 1. ‘ 𝐾 ) ) → ¬ 𝑝 ≤ 𝑊 ) |
20 |
19
|
reximi |
⊢ ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑊 ( join ‘ 𝐾 ) 𝑝 ) = ( 1. ‘ 𝐾 ) ) → ∃ 𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊 ) |
21 |
18 20
|
syl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊 ) |