Step |
Hyp |
Ref |
Expression |
1 |
|
lhpmcvr2.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lhpmcvr2.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
lhpmcvr2.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
lhpmcvr2.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
lhpmcvr2.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
lhpmcvr2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 ) |
8 |
1 2 4 7 6
|
lhpmcvr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ) |
9 |
|
simpll |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝐾 ∈ HL ) |
10 |
|
simprl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 ) |
11 |
1 6
|
lhpbase |
⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 ) |
12 |
11
|
ad2antlr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐵 ) |
13 |
1 2 3 4 7 5
|
cvrval5 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) |
14 |
9 10 12 13
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) ) |
15 |
8 14
|
mpbid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) |