Metamath Proof Explorer

Theorem islinei

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

Ref Expression
Hypotheses isline.l 
|- .<_ = ( le ` K )
isline.j 
|- .\/ = ( join ` K )
isline.a 
|- A = ( Atoms ` K )
isline.n 
|- N = ( Lines ` K )
Assertion islinei 
|- ( ( ( K e. D /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ X = { p e. A | p .<_ ( Q .\/ R ) } ) ) -> X e. N )

Proof

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