Theorem islln

Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012)

	Ref	Expression
Hypotheses	llnset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	llnset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
	llnset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	llnset.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
Assertion	islln	⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		llnset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		llnset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3		llnset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		llnset.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
5	1 2 3 4	llnset	⊢ ( 𝐾 ∈ 𝐷 → 𝑁 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } )
6	5	eleq2d	⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } ) )
7		breq2	⊢ ( 𝑥 = 𝑋 → ( 𝑝 𝐶 𝑥 ↔ 𝑝 𝐶 𝑋 ) )
8	7	rexbidv	⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 ↔ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) )
9	8	elrab	⊢ ( 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) )
10	6 9	bitrdi	⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )