Theorem llnset 34791

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

Hypotheses

Ref	Expression
llnset.b	|- B = ( Base ` K )
llnset.c	|- C = ( <o ` K )
llnset.a	|- A = ( Atoms ` K )
llnset.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
llnset	|- ( K e. D -> N = { x e. B | E. p e. A p C x } )

Distinct variable groups:   A, p   x, B   x, p, K

Allowed substitution hints:   A( x)   B( p)   C( x, p)   D( x, p)   N( x, p)

Proof of Theorem llnset

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. D -> K e. _V )
2		llnset.n	. . 3 |- N = ( LLines ` K )
3		fveq2 6191	. . . . . 6 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
4		llnset.b	. . . . . 6 |- B = ( Base ` K )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( k = K -> ( Base ` k ) = B )
6		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
7		llnset.a	. . . . . . 7 |- A = ( Atoms ` K )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Atoms ` k ) = A )
9		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( <o ` k ) = ( <o ` K ) )
10		llnset.c	. . . . . . . 8 |- C = ( <o ` K )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( <o ` k ) = C )
12	11	breqd 4664	. . . . . 6 |- ( k = K -> ( p ( <o ` k ) x <-> p C x ) )
13	8, 12	rexeqbidv 3153	. . . . 5 |- ( k = K -> ( E. p e. ( Atoms ` k ) p ( <o ` k ) x <-> E. p e. A p C x ) )
14	5, 13	rabeqbidv 3195	. . . 4 |- ( k = K -> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p ( <o ` k ) x } = { x e. B | E. p e. A p C x } )
15		df-llines 34784	. . . 4 |- LLines = ( k e. _V |-> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p ( <o ` k ) x } )
16		fvex 6201	. . . . . 6 |- ( Base ` K ) e. _V
17	4, 16	eqeltri 2697	. . . . 5 |- B e. _V
18	17	rabex 4813	. . . 4 |- { x e. B | E. p e. A p C x } e. _V
19	14, 15, 18	fvmpt 6282	. . 3 |- ( K e. _V -> ( LLines ` K ) = { x e. B | E. p e. A p C x } )
20	2, 19	syl5eq 2668	. 2 |- ( K e. _V -> N = { x e. B | E. p e. A p C x } )
21	1, 20	syl 17	1 |- ( K e. D -> N = { x e. B | E. p e. A p C x } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888   Basecbs 15857   <o ccvr 34549   Atomscatm 34550   LLinesclln 34777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-llines 34784

This theorem is referenced by:  islln  34792