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 ∧ 𝑋 ∈ 𝑃 ) → ¬ 𝑋 ∈ 𝑁 )

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	⊢ ( 𝑋 ∈ 𝑃 → 𝑋 ∈ ( 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 ∧ 𝑋 ∈ 𝑃 ) → 𝑋 ( le ‘ 𝐾 ) 𝑋 )
9	6 1 2	lplnnlelln	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑋 ∈ 𝑁 ) → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 )
10	9	3expia	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ) → ( 𝑋 ∈ 𝑁 → ¬ 𝑋 ( le ‘ 𝐾 ) 𝑋 ) )
11	8 10	mt2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ) → ¬ 𝑋 ∈ 𝑁 )