Theorem lplnnlt

Description: Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012)

	Ref	Expression
Hypotheses	lplnnlt.s	⊢ < = ( lt ‘ 𝐾 )
	lplnnlt.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
Assertion	lplnnlt	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑋 < 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		lplnnlt.s	⊢ < = ( lt ‘ 𝐾 )
2		lplnnlt.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
3	1	pltirr	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ) → ¬ 𝑋 < 𝑋 )
4	3	3adant3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑋 < 𝑋 )
5		breq2	⊢ ( 𝑋 = 𝑌 → ( 𝑋 < 𝑋 ↔ 𝑋 < 𝑌 ) )
6	5	notbid	⊢ ( 𝑋 = 𝑌 → ( ¬ 𝑋 < 𝑋 ↔ ¬ 𝑋 < 𝑌 ) )
7	4 6	syl5ibcom	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 = 𝑌 → ¬ 𝑋 < 𝑌 ) )
8		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
9	8 1	pltle	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 < 𝑌 → 𝑋 ( le ‘ 𝐾 ) 𝑌 ) )
10	8 2	lplncmp	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ↔ 𝑋 = 𝑌 ) )
11	9 10	sylibd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 < 𝑌 → 𝑋 = 𝑌 ) )
12	11	necon3ad	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 ≠ 𝑌 → ¬ 𝑋 < 𝑌 ) )
13	7 12	pm2.61dne	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑋 < 𝑌 )