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Theorem brcolinear2 25240
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 25239 . . . 4  |-  Rel  Colinear
21brrelexi 4811 . . 3  |-  ( P 
Colinear 
<. Q ,  R >.  ->  P  e.  _V )
32a1i 10 . 2  |-  ( ( Q  e.  V  /\  R  e.  W )  ->  ( P  Colinear  <. Q ,  R >.  ->  P  e.  _V ) )
4 elex 2872 . . . . . 6  |-  ( P  e.  ( EE `  n )  ->  P  e.  _V )
543ad2ant1 976 . . . . 5  |-  ( ( P  e.  ( EE
`  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n
) )  ->  P  e.  _V )
65adantr 451 . . . 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 2739 . . 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 10 . 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 4105 . . . . . 6  |-  ( P 
Colinear 
<. Q ,  R >.  <->  <. P ,  <. Q ,  R >. >.  e.  Colinear  )
10 df-colinear 25223 . . . . . . 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 2422 . . . . . 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 240 . . . . 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 4319 . . . . . . 7  |-  <. Q ,  R >.  e.  _V
14 opelcnvg 4943 . . . . . . 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 652 . . . . . 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 978 . . . . 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 248 . . . 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 2418 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  e.  ( EE
`  n )  <->  Q  e.  ( EE `  n ) ) )
19183anbi2d 1257 . . . . . . 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 3877 . . . . . . . . 9  |-  ( q  =  Q  ->  <. q ,  r >.  =  <. Q ,  r >. )
2120breq2d 4116 . . . . . . . 8  |-  ( q  =  Q  ->  (
p  Btwn  <. q ,  r >.  <->  p  Btwn  <. Q , 
r >. ) )
22 breq1 4107 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  Btwn  <. r ,  p >.  <->  Q  Btwn  <. r ,  p >. ) )
23 opeq2 3878 . . . . . . . . 9  |-  ( q  =  Q  ->  <. p ,  q >.  =  <. p ,  Q >. )
2423breq2d 4116 . . . . . . . 8  |-  ( q  =  Q  ->  (
r  Btwn  <. p ,  q >.  <->  r  Btwn  <. p ,  Q >. ) )
2521, 22, 243orbi123d 1251 . . . . . . 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 691 . . . . . 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 2640 . . . . 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 2418 . . . . . . . 8  |-  ( r  =  R  ->  (
r  e.  ( EE
`  n )  <->  R  e.  ( EE `  n ) ) )
29283anbi3d 1258 . . . . . . 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 3878 . . . . . . . . 9  |-  ( r  =  R  ->  <. Q , 
r >.  =  <. Q ,  R >. )
3130breq2d 4116 . . . . . . . 8  |-  ( r  =  R  ->  (
p  Btwn  <. Q , 
r >. 
<->  p  Btwn  <. Q ,  R >. ) )
32 opeq1 3877 . . . . . . . . 9  |-  ( r  =  R  ->  <. r ,  p >.  =  <. R ,  p >. )
3332breq2d 4116 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  Btwn  <. r ,  p >.  <-> 
Q  Btwn  <. R ,  p >. ) )
34 breq1 4107 . . . . . . . 8  |-  ( r  =  R  ->  (
r  Btwn  <. p ,  Q >.  <->  R  Btwn  <. p ,  Q >. ) )
3531, 33, 343orbi123d 1251 . . . . . . 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 691 . . . . . 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 2640 . . . . 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 2418 . . . . . . . 8  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
39383anbi1d 1256 . . . . . . 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 4107 . . . . . . . 8  |-  ( p  =  P  ->  (
p  Btwn  <. Q ,  R >. 
<->  P  Btwn  <. Q ,  R >. ) )
41 opeq2 3878 . . . . . . . . 9  |-  ( p  =  P  ->  <. R ,  p >.  =  <. R ,  P >. )
4241breq2d 4116 . . . . . . . 8  |-  ( p  =  P  ->  ( Q  Btwn  <. R ,  p >.  <-> 
Q  Btwn  <. R ,  P >. ) )
43 opeq1 3877 . . . . . . . . 9  |-  ( p  =  P  ->  <. p ,  Q >.  =  <. P ,  Q >. )
4443breq2d 4116 . . . . . . . 8  |-  ( p  =  P  ->  ( R  Btwn  <. p ,  Q >.  <-> 
R  Btwn  <. P ,  Q >. ) )
4540, 42, 443orbi123d 1251 . . . . . . 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 691 . . . . . 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 2640 . . . . 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 6022 . . . 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 244 . . 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 1153 . 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 343 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 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   <.cop 3719   class class class wbr 4104   `'ccnv 4770   ` cfv 5337   {coprab 5946   NNcn 9836   EEcee 25075    Btwn cbtwn 25076    Colinear ccolin 25219
This theorem is referenced by:  brcolinear  25241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-oprab 5949  df-colinear 25223
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