Metamath Proof Explorer


Theorem lplnnelln

Description: No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012)

Ref Expression
Hypotheses lplnnelln.n 𝑁 = ( LLines ‘ 𝐾 )
lplnnelln.p 𝑃 = ( LPlanes ‘ 𝐾 )
Assertion lplnnelln ( ( 𝐾 ∈ HL ∧ 𝑋𝑃 ) → ¬ 𝑋𝑁 )

Proof

Step Hyp Ref Expression
1 lplnnelln.n 𝑁 = ( LLines ‘ 𝐾 )
2 lplnnelln.p 𝑃 = ( LPlanes ‘ 𝐾 )
3 hllat ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5 4 2 lplnbase ( 𝑋𝑃𝑋 ∈ ( Base ‘ 𝐾 ) )
6 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
7 4 6 latref ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
8 3 5 7 syl2an ( ( 𝐾 ∈ HL ∧ 𝑋𝑃 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
9 6 1 2 lplnnlelln ( ( 𝐾 ∈ HL ∧ 𝑋𝑃𝑋𝑁 ) → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 )
10 9 3expia ( ( 𝐾 ∈ HL ∧ 𝑋𝑃 ) → ( 𝑋𝑁 → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) )
11 8 10 mt2d ( ( 𝐾 ∈ HL ∧ 𝑋𝑃 ) → ¬ 𝑋𝑁 )