Theorem islln2

Description: The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012)

	Ref	Expression
Hypotheses	islln3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	islln3.j	⊢ ∨ = ( join ‘ 𝐾 )
	islln3.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	islln3.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
Assertion	islln2	⊢ ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		islln3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		islln3.j	⊢ ∨ = ( join ‘ 𝐾 )
3		islln3.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		islln3.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
5	1 4	llnbase	⊢ ( 𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵 )
6	5	pm4.71ri	⊢ ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁 ) )
7	1 2 3 4	islln3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑁 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) )
8	7	pm5.32da	⊢ ( 𝐾 ∈ HL → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) ) )
9	6 8	bitrid	⊢ ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) ) )