Theorem 3noncolr1N

Description: Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	3noncol.l	⊢ ≤ = ( le ‘ 𝐾 )
	3noncol.j	⊢ ∨ = ( join ‘ 𝐾 )
	3noncol.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	3noncolr1N	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ≠ 𝑃 ∧ ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3noncol.l	⊢ ≤ = ( le ‘ 𝐾 )
2		3noncol.j	⊢ ∨ = ( join ‘ 𝐾 )
3		3noncol.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		simp1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
5		simp22	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
6		simp23	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑅 ∈ 𝐴 )
7		simp21	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
8	1 2 3	3noncolr2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) )
9	1 2 3	3noncolr2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑅 ≠ 𝑃 ∧ ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ) )
10	4 5 6 7 8 9	syl131anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ≠ 𝑃 ∧ ¬ 𝑄 ≤ ( 𝑅 ∨ 𝑃 ) ) )