Theorem lplnset

Description: The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012)

	Ref	Expression
Hypotheses	lplnset.b	|- B = ( Base ` K )
	lplnset.c	|- C = (
	lplnset.n	|- N = ( LLines ` K )
	lplnset.p	|- P = ( LPlanes ` K )
Assertion	lplnset	|- ( K e. A -> P = { x e. B | E. y e. N y C x } )

Proof

Step	Hyp	Ref	Expression
1		lplnset.b	|- B = ( Base ` K )
2		lplnset.c	|- C = (
3		lplnset.n	|- N = ( LLines ` K )
4		lplnset.p	|- P = ( LPlanes ` K )
5		elex	|- ( K e. A -> K e. _V )
6		fveq2	|- ( k = K -> ( Base ` k ) = ( Base ` K ) )
7	6 1	eqtr4di	|- ( k = K -> ( Base ` k ) = B )
8		fveq2	|- ( k = K -> ( LLines ` k ) = ( LLines ` K ) )
9	8 3	eqtr4di	|- ( k = K -> ( LLines ` k ) = N )
10		fveq2	|- ( k = K -> (
11	10 2	eqtr4di	|- ( k = K -> (
12	11	breqd	|- ( k = K -> ( y ( y C x ) )
13	9 12	rexeqbidv	|- ( k = K -> ( E. y e. ( LLines ` k ) y ( E. y e. N y C x ) )
14	7 13	rabeqbidv	|- ( k = K -> { x e. ( Base ` k ) | E. y e. ( LLines ` k ) y (
15		df-lplanes	|- LPlanes = ( k e. _V |-> { x e. ( Base ` k ) | E. y e. ( LLines ` k ) y (
16	1	fvexi	|- B e. _V
17	16	rabex	|- { x e. B | E. y e. N y C x } e. _V
18	14 15 17	fvmpt	|- ( K e. _V -> ( LPlanes ` K ) = { x e. B | E. y e. N y C x } )
19	4 18	eqtrid	|- ( K e. _V -> P = { x e. B | E. y e. N y C x } )
20	5 19	syl	|- ( K e. A -> P = { x e. B | E. y e. N y C x } )