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 = (
	llnset.a	|- A = ( Atoms ` K )
	llnset.n	|- N = ( LLines ` K )
Assertion	llnset	|- ( K e. D -> N = { x e. B | E. p e. A p C x } )

Proof

Step	Hyp	Ref	Expression
1		llnset.b	|- B = ( Base ` K )
2		llnset.c	|- C = (
3		llnset.a	|- A = ( Atoms ` K )
4		llnset.n	|- N = ( LLines ` K )
5		elex	|- ( K e. D -> K e. _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 -> (
11	10 2	eqtr4di	|- ( k = K -> (
12	11	breqd	|- ( k = K -> ( p ( p C x ) )
13	9 12	rexeqbidv	|- ( k = K -> ( E. p e. ( Atoms ` k ) p ( E. p e. A p C x ) )
14	7 13	rabeqbidv	|- ( k = K -> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p (
15		df-llines	|- LLines = ( k e. _V |-> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p (
16	1	fvexi	|- B e. _V
17	16	rabex	|- { x e. B | E. p e. A p C x } e. _V
18	14 15 17	fvmpt	|- ( K e. _V -> ( LLines ` K ) = { x e. B | E. p e. A p C x } )
19	4 18	syl5eq	|- ( K e. _V -> N = { x e. B | E. p e. A p C x } )
20	5 19	syl	|- ( K e. D -> N = { x e. B | E. p e. A p C x } )