Step |
Hyp |
Ref |
Expression |
1 |
|
lhpatltex.s |
⊢ < = ( lt ‘ 𝐾 ) |
2 |
|
lhpatltex.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
lhpatltex.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
simpl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ HL ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
6 |
5 3
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) ) |
8 |
|
eqid |
⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
10 |
8 9 3
|
lhp1cvr |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) |
11 |
5 1 8 9 2
|
1cvratex |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) → ∃ 𝑝 ∈ 𝐴 𝑝 < 𝑊 ) |
12 |
4 7 10 11
|
syl3anc |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 𝑝 < 𝑊 ) |