Theorem islinei 35026

Description: Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)

Hypotheses

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

Assertion

Ref	Expression
islinei	|- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> X e. N )

Distinct variable groups:   A, p   K, p   Q, p   R, p

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

Proof of Theorem islinei

Dummy variables q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . 3 |- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> Q e. A )
2		simpl3 1066	. . 3 |- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> R e. A )
3		simpr 477	. . 3 |- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) )
4		neeq1 2856	. . . . 5 |- ( q = Q -> ( q =/= r <-> Q =/= r ) )
5		oveq1 6657	. . . . . . . 8 |- ( q = Q -> ( q .\/ r ) = ( Q .\/ r ) )
6	5	breq2d 4665	. . . . . . 7 |- ( q = Q -> ( p .<_ ( q .\/ r ) <-> p .<_ ( Q .\/ r ) ) )
7	6	rabbidv 3189	. . . . . 6 |- ( q = Q -> { p e. A | p .<_ ( q .\/ r ) } = { p e. A | p .<_ ( Q .\/ r ) } )
8	7	eqeq2d 2632	. . . . 5 |- ( q = Q -> ( X = { p e. A | p .<_ ( q .\/ r ) } <-> X = { p e. A | p .<_ ( Q .\/ r ) } ) )
9	4, 8	anbi12d 747	. . . 4 |- ( q = Q -> ( ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) <-> ( Q =/= r /\ X = { p e. A | p .<_ ( Q .\/ r ) } ) ) )
10		neeq2 2857	. . . . 5 |- ( r = R -> ( Q =/= r <-> Q =/= R ) )
11		oveq2 6658	. . . . . . . 8 |- ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) )
12	11	breq2d 4665	. . . . . . 7 |- ( r = R -> ( p .<_ ( Q .\/ r ) <-> p .<_ ( Q .\/ R ) ) )
13	12	rabbidv 3189	. . . . . 6 |- ( r = R -> { p e. A | p .<_ ( Q .\/ r ) } = { p e. A | p .<_ ( Q .\/ R ) } )
14	13	eqeq2d 2632	. . . . 5 |- ( r = R -> ( X = { p e. A | p .<_ ( Q .\/ r ) } <-> X = { p e. A | p .<_ ( Q .\/ R ) } ) )
15	10, 14	anbi12d 747	. . . 4 |- ( r = R -> ( ( Q =/= r /\ X = { p e. A | p .<_ ( Q .\/ r ) } ) <-> ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) )
16	9, 15	rspc2ev 3324	. . 3 |- ( ( Q e. A /\ 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 ) } ) )
17	1, 2, 3, 16	syl3anc 1326	. 2 |- ( ( ( K e. D /\ Q e. A /\ 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		simpl1 1064	. . 3 |- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> K e. D )
19		isline.l	. . . 4 |- .<_ = ( le ` K )
20		isline.j	. . . 4 |- .\/ = ( join ` K )
21		isline.a	. . . 4 |- A = ( Atoms ` K )
22		isline.n	. . . 4 |- N = ( Lines ` K )
23	19, 20, 21, 22	isline 35025	. . 3 |- ( K e. D -> ( X e. N <-> E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )
24	18, 23	syl 17	. 2 |- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> ( X e. N <-> E. q e. A E. r e. A ( q =/= r /\ X = { p e. A | p .<_ ( q .\/ r ) } ) ) )
25	17, 24	mpbird 247	1 |- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> X e. N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   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:  linepmap  35061