Theorem isline 35025

Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)

Hypotheses

Ref	Expression
isline.l	|- .<_ = ( le ` K )
isline.j	|- .\/ = ( join ` K )
isline.a	|- A = ( Atoms ` K )
isline.n	|- N = ( Lines ` K )

Assertion

Ref	Expression
isline	|- ( K e. D -> ( X e. N <-> E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )

Distinct variable groups:   q, p, r, A   K, p, q, r   X, q, r

Allowed substitution hints:   D( r, q, p)   .\/ ( r, q, p)   .<_ ( r, q, p)   N( r, q, p)   X( p)

Proof of Theorem isline

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isline.l	. . . 4 |- .<_ = ( le ` K )
2		isline.j	. . . 4 |- .\/ = ( join ` K )
3		isline.a	. . . 4 |- A = ( Atoms ` K )
4		isline.n	. . . 4 |- N = ( Lines ` K )
5	1, 2, 3, 4	lineset 35024	. . 3 |- ( K e. D -> N = { x | E. q e. A E. r e. A ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) } )
6	5	eleq2d 2687	. 2 |- ( K e. D -> ( X e. N <-> X e. { x | E. q e. A E. r e. A ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) } ) )
7		fvex 6201	. . . . . . . . 9 |- ( Atoms ` K ) e. _V
8	3, 7	eqeltri 2697	. . . . . . . 8 |- A e. _V
9	8	rabex 4813	. . . . . . 7 |- { p e. A | p .<_ ( q .\/ r ) } e. _V
10		eleq1 2689	. . . . . . 7 |- ( X = { p e. A | p .<_ ( q .\/ r ) } -> ( X e. _V <-> { p e. A | p .<_ ( q .\/ r ) } e. _V ) )
11	9, 10	mpbiri 248	. . . . . 6 |- ( X = { p e. A | p .<_ ( q .\/ r ) } -> X e. _V )
12	11	adantl 482	. . . . 5 |- ( ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) -> X e. _V )
13	12	a1i 11	. . . 4 |- ( ( q e. A /\ r e. A ) -> ( ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) -> X e. _V ) )
14	13	rexlimivv 3036	. . 3 |- ( E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) -> X e. _V )
15		eqeq1 2626	. . . . 5 |- ( x = X -> ( x = { p e. A | p .<_ ( q .\/ r ) } <-> X = { p e. A | p .<_ ( q .\/ r ) } ) )
16	15	anbi2d 740	. . . 4 |- ( x = X -> ( ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) <-> ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )
17	16	2rexbidv 3057	. . 3 |- ( x = X -> ( E. q e. A E. r e. A ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) <-> E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )
18	14, 17	elab3 3358	. 2 |- ( X e. { x | E. q e. A E. r e. A ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) } <-> E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) )
19	6, 18	syl6bb 276	1 |- ( K e. D -> ( X e. N <-> E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   Linesclines 34780

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lines 34787

This theorem is referenced by:  islinei  35026  linepsubN  35038  isline2  35060