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Theorem brcolinear2 25904
Description: Alternate colinearity binary relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
brcolinear2  |-  ( ( Q  e.  V  /\  R  e.  W )  ->  ( P  Colinear  <. Q ,  R >. 
<->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
Distinct variable groups:    P, n    Q, n    R, n
Allowed substitution hints:    V( n)    W( n)

Proof of Theorem brcolinear2
Dummy variables  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 colinrel 25903 . . . 4  |-  Rel  Colinear
21brrelexi 4885 . . 3  |-  ( P 
Colinear 
<. Q ,  R >.  ->  P  e.  _V )
32a1i 11 . 2  |-  ( ( Q  e.  V  /\  R  e.  W )  ->  ( P  Colinear  <. Q ,  R >.  ->  P  e.  _V ) )
4 elex 2932 . . . . . 6  |-  ( P  e.  ( EE `  n )  ->  P  e.  _V )
543ad2ant1 978 . . . . 5  |-  ( ( P  e.  ( EE
`  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n
) )  ->  P  e.  _V )
65adantr 452 . . . 4  |-  ( ( ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) )  ->  P  e.  _V )
76rexlimivw 2794 . . 3  |-  ( E. n  e.  NN  (
( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) )  ->  P  e.  _V )
87a1i 11 . 2  |-  ( ( Q  e.  V  /\  R  e.  W )  ->  ( E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
)  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) )  ->  P  e.  _V ) )
9 df-br 4181 . . . . . 6  |-  ( P 
Colinear 
<. Q ,  R >.  <->  <. P ,  <. Q ,  R >. >.  e.  Colinear  )
10 df-colinear 25887 . . . . . . 7  |-  Colinear  =  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }
1110eleq2i 2476 . . . . . 6  |-  ( <. P ,  <. Q ,  R >. >.  e.  Colinear  <->  <. P ,  <. Q ,  R >. >.  e.  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } )
129, 11bitri 241 . . . . 5  |-  ( P 
Colinear 
<. Q ,  R >.  <->  <. P ,  <. Q ,  R >. >.  e.  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } )
13 opex 4395 . . . . . . 7  |-  <. Q ,  R >.  e.  _V
14 opelcnvg 5019 . . . . . . 7  |-  ( ( P  e.  _V  /\  <. Q ,  R >.  e. 
_V )  ->  ( <. P ,  <. Q ,  R >. >.  e.  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }  <->  <. <. Q ,  R >. ,  P >.  e. 
{ <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } ) )
1513, 14mpan2 653 . . . . . 6  |-  ( P  e.  _V  ->  ( <. P ,  <. Q ,  R >. >.  e.  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }  <->  <. <. Q ,  R >. ,  P >.  e. 
{ <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } ) )
16153ad2ant3 980 . . . . 5  |-  ( ( Q  e.  V  /\  R  e.  W  /\  P  e.  _V )  ->  ( <. P ,  <. Q ,  R >. >.  e.  `' { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }  <->  <. <. Q ,  R >. ,  P >.  e. 
{ <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } ) )
1712, 16syl5bb 249 . . . 4  |-  ( ( Q  e.  V  /\  R  e.  W  /\  P  e.  _V )  ->  ( P  Colinear  <. Q ,  R >. 
<-> 
<. <. Q ,  R >. ,  P >.  e.  { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) } ) )
18 eleq1 2472 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  e.  ( EE
`  n )  <->  Q  e.  ( EE `  n ) ) )
19183anbi2d 1259 . . . . . . 7  |-  ( q  =  Q  ->  (
( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  <->  ( p  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) ) ) )
20 opeq1 3952 . . . . . . . . 9  |-  ( q  =  Q  ->  <. q ,  r >.  =  <. Q ,  r >. )
2120breq2d 4192 . . . . . . . 8  |-  ( q  =  Q  ->  (
p  Btwn  <. q ,  r >.  <->  p  Btwn  <. Q , 
r >. ) )
22 breq1 4183 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  Btwn  <. r ,  p >.  <->  Q  Btwn  <. r ,  p >. ) )
23 opeq2 3953 . . . . . . . . 9  |-  ( q  =  Q  ->  <. p ,  q >.  =  <. p ,  Q >. )
2423breq2d 4192 . . . . . . . 8  |-  ( q  =  Q  ->  (
r  Btwn  <. p ,  q >.  <->  r  Btwn  <. p ,  Q >. ) )
2521, 22, 243orbi123d 1253 . . . . . . 7  |-  ( q  =  Q  ->  (
( p  Btwn  <. q ,  r >.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q >. )  <->  ( p  Btwn  <. Q , 
r >.  \/  Q  Btwn  <.
r ,  p >.  \/  r  Btwn  <. p ,  Q >. ) ) )
2619, 25anbi12d 692 . . . . . 6  |-  ( q  =  Q  ->  (
( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) )  <->  ( (
p  e.  ( EE
`  n )  /\  Q  e.  ( EE `  n )  /\  r  e.  ( EE `  n
) )  /\  (
p  Btwn  <. Q , 
r >.  \/  Q  Btwn  <.
r ,  p >.  \/  r  Btwn  <. p ,  Q >. ) ) ) )
2726rexbidv 2695 . . . . 5  |-  ( q  =  Q  ->  ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) )  <->  E. n  e.  NN  ( ( p  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
)  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. Q ,  r
>.  \/  Q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  Q >. ) ) ) )
28 eleq1 2472 . . . . . . . 8  |-  ( r  =  R  ->  (
r  e.  ( EE
`  n )  <->  R  e.  ( EE `  n ) ) )
29283anbi3d 1260 . . . . . . 7  |-  ( r  =  R  ->  (
( p  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  <->  ( p  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) ) ) )
30 opeq2 3953 . . . . . . . . 9  |-  ( r  =  R  ->  <. Q , 
r >.  =  <. Q ,  R >. )
3130breq2d 4192 . . . . . . . 8  |-  ( r  =  R  ->  (
p  Btwn  <. Q , 
r >. 
<->  p  Btwn  <. Q ,  R >. ) )
32 opeq1 3952 . . . . . . . . 9  |-  ( r  =  R  ->  <. r ,  p >.  =  <. R ,  p >. )
3332breq2d 4192 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  Btwn  <. r ,  p >.  <-> 
Q  Btwn  <. R ,  p >. ) )
34 breq1 4183 . . . . . . . 8  |-  ( r  =  R  ->  (
r  Btwn  <. p ,  Q >.  <->  R  Btwn  <. p ,  Q >. ) )
3531, 33, 343orbi123d 1253 . . . . . . 7  |-  ( r  =  R  ->  (
( p  Btwn  <. Q , 
r >.  \/  Q  Btwn  <.
r ,  p >.  \/  r  Btwn  <. p ,  Q >. )  <->  ( p  Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  p >.  \/  R  Btwn  <.
p ,  Q >. ) ) )
3629, 35anbi12d 692 . . . . . 6  |-  ( r  =  R  ->  (
( ( p  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. Q ,  r
>.  \/  Q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  Q >. ) )  <->  ( (
p  e.  ( EE
`  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n
) )  /\  (
p  Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  p >.  \/  R  Btwn  <. p ,  Q >. ) ) ) )
3736rexbidv 2695 . . . . 5  |-  ( r  =  R  ->  ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. Q ,  r
>.  \/  Q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  Q >. ) )  <->  E. n  e.  NN  ( ( p  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
)  /\  R  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  p >.  \/  R  Btwn  <.
p ,  Q >. ) ) ) )
38 eleq1 2472 . . . . . . . 8  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
39383anbi1d 1258 . . . . . . 7  |-  ( p  =  P  ->  (
( p  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  <->  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) ) ) )
40 breq1 4183 . . . . . . . 8  |-  ( p  =  P  ->  (
p  Btwn  <. Q ,  R >. 
<->  P  Btwn  <. Q ,  R >. ) )
41 opeq2 3953 . . . . . . . . 9  |-  ( p  =  P  ->  <. R ,  p >.  =  <. R ,  P >. )
4241breq2d 4192 . . . . . . . 8  |-  ( p  =  P  ->  ( Q  Btwn  <. R ,  p >.  <-> 
Q  Btwn  <. R ,  P >. ) )
43 opeq1 3952 . . . . . . . . 9  |-  ( p  =  P  ->  <. p ,  Q >.  =  <. P ,  Q >. )
4443breq2d 4192 . . . . . . . 8  |-  ( p  =  P  ->  ( R  Btwn  <. p ,  Q >.  <-> 
R  Btwn  <. P ,  Q >. ) )
4540, 42, 443orbi123d 1253 . . . . . . 7  |-  ( p  =  P  ->  (
( p  Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  p >.  \/  R  Btwn  <. p ,  Q >. )  <->  ( P  Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) )
4639, 45anbi12d 692 . . . . . 6  |-  ( p  =  P  ->  (
( ( p  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  p >.  \/  R  Btwn  <.
p ,  Q >. ) )  <->  ( ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
)  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
4746rexbidv 2695 . . . . 5  |-  ( p  =  P  ->  ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  p >.  \/  R  Btwn  <.
p ,  Q >. ) )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
4827, 37, 47eloprabg 6128 . . . 4  |-  ( ( Q  e.  V  /\  R  e.  W  /\  P  e.  _V )  ->  ( <. <. Q ,  R >. ,  P >.  e.  { <. <. q ,  r
>. ,  p >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  r  e.  ( EE `  n ) )  /\  ( p 
Btwn  <. q ,  r
>.  \/  q  Btwn  <. r ,  p >.  \/  r  Btwn  <. p ,  q
>. ) ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
)  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
4917, 48bitrd 245 . . 3  |-  ( ( Q  e.  V  /\  R  e.  W  /\  P  e.  _V )  ->  ( P  Colinear  <. Q ,  R >. 
<->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
50493expia 1155 . 2  |-  ( ( Q  e.  V  /\  R  e.  W )  ->  ( P  e.  _V  ->  ( P  Colinear  <. Q ,  R >. 
<->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) ) )
513, 8, 50pm5.21ndd 344 1  |-  ( ( Q  e.  V  /\  R  e.  W )  ->  ( P  Colinear  <. Q ,  R >. 
<->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n ) )  /\  ( P 
Btwn  <. Q ,  R >.  \/  Q  Btwn  <. R ,  P >.  \/  R  Btwn  <. P ,  Q >. ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2675   _Vcvv 2924   <.cop 3785   class class class wbr 4180   `'ccnv 4844   ` cfv 5421   {coprab 6049   NNcn 9964   EEcee 25739    Btwn cbtwn 25740    Colinear ccolin 25883
This theorem is referenced by:  brcolinear  25905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-oprab 6052  df-colinear 25887
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