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Theorem brcolinear 29677
Description: The binary relationship form of the colinearity predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
brcolinear  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )

Proof of Theorem brcolinear
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 brcolinear2 29676 . . . 4  |-  ( ( B  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
213adant1 1013 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  ( A  Colinear  <. B ,  C >.  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
32adantl 466 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
4 simpr 461 . . . 4  |-  ( ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )  ->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )
54rexlimivw 2930 . . 3  |-  ( E. n  e.  NN  (
( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )  ->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )
6 fveq2 5852 . . . . . . . 8  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
76eleq2d 2511 . . . . . . 7  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
86eleq2d 2511 . . . . . . 7  |-  ( n  =  N  ->  ( B  e.  ( EE `  n )  <->  B  e.  ( EE `  N ) ) )
96eleq2d 2511 . . . . . . 7  |-  ( n  =  N  ->  ( C  e.  ( EE `  n )  <->  C  e.  ( EE `  N ) ) )
107, 8, 93anbi123d 1298 . . . . . 6  |-  ( n  =  N  ->  (
( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) ) )
1110anbi1d 704 . . . . 5  |-  ( n  =  N  ->  (
( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )  <->  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
1211rspcev 3194 . . . 4  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )  ->  E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
1312expr 615 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  ->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
145, 13impbid2 204 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) )  <->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
153, 14bitrd 253 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 971    /\ w3a 972    = wceq 1381    e. wcel 1802   E.wrex 2792   <.cop 4016   class class class wbr 4433   ` cfv 5574   NNcn 10537   EEcee 24056    Btwn cbtwn 24057    Colinear ccolin 29655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-cnv 4993  df-iota 5537  df-fv 5582  df-oprab 6281  df-colinear 29657
This theorem is referenced by:  colinearperm1  29680  colinearperm3  29681  colineartriv1  29685  colineartriv2  29686  btwncolinear1  29687  colinearxfr  29693  lineext  29694  fscgr  29698  colinbtwnle  29736  broutsideof2  29740  lineunray  29765  lineelsb2  29766
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