Theorem isline 35876

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 35875	. . 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 2452	. 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 5784	. . . . . . . . 9 |- ( Atoms ` K ) e. _V
8	3, 7	eqeltri 2466	. . . . . . . 8 |- A e. _V
9	8	rabex 4516	. . . . . . 7 |- { p e. A | p .<_ ( q .\/ r ) } e. _V
10		eleq1 2454	. . . . . . 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 464	. . . . 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 2879	. . 3 |- ( E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) -> X e. _V )
15		eqeq1 2386	. . . . 5 |- ( x = X -> ( x = { p e. A | p .<_ ( q .\/ r ) } <-> X = { p e. A | p .<_ ( q .\/ r ) } ) )
16	15	anbi2d 701	. . . 4 |- ( x = X -> ( ( q =/= r /\ x = { p e. A | p .<_ ( q .\/ r ) } ) <-> ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )
17	16	2rexbidv 2900	. . 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 3178	. 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 367   = wceq 1399   e. wcel 1826   {cab 2367   =/= wne 2577   E.wrex 2733   {crab 2736   _Vcvv 3034   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   lecple 14709   joincjn 15690   Atomscatm 35401   Linesclines 35631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-lines 35638

This theorem is referenced by:  islinei  35877  linepsubN  35889  isline2  35911