Theorem llnset

Description: The set of lattice lines in a Hilbert lattice. (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	llnset	⊢ ( 𝐾 ∈ 𝐷 → 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		elex	⊢ ( 𝐾 ∈ 𝐷 → 𝐾 ∈ V )
6		fveq2	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) )
7	6 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = 𝐵 )
8		fveq2	⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = ( Atoms ‘ 𝐾 ) )
9	8 3	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = 𝐴 )
10		fveq2	⊢ ( 𝑘 = 𝐾 → ( ⋖ ‘ 𝑘 ) = ( ⋖ ‘ 𝐾 ) )
11	10 2	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( ⋖ ‘ 𝑘 ) = 𝐶 )
12	11	breqd	⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( ⋖ ‘ 𝑘 ) 𝑥 ↔ 𝑝 𝐶 𝑥 ) )
13	9 12	rexeqbidv	⊢ ( 𝑘 = 𝐾 → ( ∃ 𝑝 ∈ ( Atoms ‘ 𝑘 ) 𝑝 ( ⋖ ‘ 𝑘 ) 𝑥 ↔ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 ) )
14	7 13	rabeqbidv	⊢ ( 𝑘 = 𝐾 → { 𝑥 ∈ ( Base ‘ 𝑘 ) ∣ ∃ 𝑝 ∈ ( Atoms ‘ 𝑘 ) 𝑝 ( ⋖ ‘ 𝑘 ) 𝑥 } = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } )
15		df-llines	⊢ LLines = ( 𝑘 ∈ V ↦ { 𝑥 ∈ ( Base ‘ 𝑘 ) ∣ ∃ 𝑝 ∈ ( Atoms ‘ 𝑘 ) 𝑝 ( ⋖ ‘ 𝑘 ) 𝑥 } )
16	1	fvexi	⊢ 𝐵 ∈ V
17	16	rabex	⊢ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } ∈ V
18	14 15 17	fvmpt	⊢ ( 𝐾 ∈ V → ( LLines ‘ 𝐾 ) = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } )
19	4 18	eqtrid	⊢ ( 𝐾 ∈ V → 𝑁 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } )
20	5 19	syl	⊢ ( 𝐾 ∈ 𝐷 → 𝑁 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } )