Theorem isline 33665

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 33664	. . 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 2519	. 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 5785	. . . . . . . . 9 |- ( Atoms ` K ) e. _V
8	3, 7	eqeltri 2532	. . . . . . . 8 |- A e. _V
9	8	rabex 4527	. . . . . . 7 |- { p e. A | p .<_ ( q .\/ r ) } e. _V
10		eleq1 2520	. . . . . . 7 |- ( X = { p e. A | p .<_ ( q .\/ r ) } -> ( X e. _V <-> { p e. A | p .<_ ( q .\/ r ) } e. _V ) )
11	9, 10	mpbiri 233	. . . . . 6 |- ( X = { p e. A | p .<_ ( q .\/ r ) } -> X e. _V )
12	11	adantl 466	. . . . 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 2928	. . 3 |- ( E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) -> X e. _V )
15		eqeq1 2453	. . . . 5 |- ( x = X -> ( x = { p e. A | p .<_ ( q .\/ r ) } <-> X = { p e. A | p .<_ ( q .\/ r ) } ) )
16	15	anbi2d 703	. . . 4 |- ( x = X -> ( ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) <-> ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )
17	16	2rexbidv 2839	. . 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 3196	. 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 261	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 184   /\ wa 369   = wceq 1370   e. wcel 1757   {cab 2435   =/= wne 2641   E.wrex 2793   {crab 2796   _Vcvv 3054   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   lecple 14333   joincjn 15202   Atomscatm 33190   Linesclines 33420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pr 4615  ax-un 6458

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-lines 33427

This theorem is referenced by:  islinei  33666  linepsubN  33678  isline2  33700