Metamath Proof Explorer
Description: The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012)
|
|
Ref |
Expression |
|
Hypotheses |
lplnset.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
|
|
lplnset.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
|
|
lplnset.n |
⊢ 𝑁 = ( LLines ‘ 𝐾 ) |
|
|
lplnset.p |
⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) |
|
Assertion |
islpln4 |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑦 ∈ 𝑁 𝑦 𝐶 𝑋 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lplnset.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lplnset.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
3 |
|
lplnset.n |
⊢ 𝑁 = ( LLines ‘ 𝐾 ) |
4 |
|
lplnset.p |
⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) |
5 |
1 2 3 4
|
islpln |
⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑃 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑁 𝑦 𝐶 𝑋 ) ) ) |
6 |
5
|
baibd |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑦 ∈ 𝑁 𝑦 𝐶 𝑋 ) ) |