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Theorem brcolinear2 29561
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 29560 . . . 4  |-  Rel  Colinear
21brrelexi 5040 . . 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 3122 . . . . . 6  |-  ( P  e.  ( EE `  n )  ->  P  e.  _V )
543ad2ant1 1017 . . . . 5  |-  ( ( P  e.  ( EE
`  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n
) )  ->  P  e.  _V )
65adantr 465 . . . 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 2952 . . 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 4448 . . . . . 6  |-  ( P 
Colinear 
<. Q ,  R >.  <->  <. P ,  <. Q ,  R >. >.  e.  Colinear  )
10 df-colinear 29542 . . . . . . 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 2545 . . . . . 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 249 . . . . 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 4711 . . . . . . 7  |-  <. Q ,  R >.  e.  _V
14 opelcnvg 5182 . . . . . . 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 671 . . . . . 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 1019 . . . . 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 257 . . . 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 2539 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  e.  ( EE
`  n )  <->  Q  e.  ( EE `  n ) ) )
19183anbi2d 1304 . . . . . . 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 4213 . . . . . . . . 9  |-  ( q  =  Q  ->  <. q ,  r >.  =  <. Q ,  r >. )
2120breq2d 4459 . . . . . . . 8  |-  ( q  =  Q  ->  (
p  Btwn  <. q ,  r >.  <->  p  Btwn  <. Q , 
r >. ) )
22 breq1 4450 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  Btwn  <. r ,  p >.  <->  Q  Btwn  <. r ,  p >. ) )
23 opeq2 4214 . . . . . . . . 9  |-  ( q  =  Q  ->  <. p ,  q >.  =  <. p ,  Q >. )
2423breq2d 4459 . . . . . . . 8  |-  ( q  =  Q  ->  (
r  Btwn  <. p ,  q >.  <->  r  Btwn  <. p ,  Q >. ) )
2521, 22, 243orbi123d 1298 . . . . . . 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 710 . . . . . 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 2973 . . . . 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 2539 . . . . . . . 8  |-  ( r  =  R  ->  (
r  e.  ( EE
`  n )  <->  R  e.  ( EE `  n ) ) )
29283anbi3d 1305 . . . . . . 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 4214 . . . . . . . . 9  |-  ( r  =  R  ->  <. Q , 
r >.  =  <. Q ,  R >. )
3130breq2d 4459 . . . . . . . 8  |-  ( r  =  R  ->  (
p  Btwn  <. Q , 
r >. 
<->  p  Btwn  <. Q ,  R >. ) )
32 opeq1 4213 . . . . . . . . 9  |-  ( r  =  R  ->  <. r ,  p >.  =  <. R ,  p >. )
3332breq2d 4459 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  Btwn  <. r ,  p >.  <-> 
Q  Btwn  <. R ,  p >. ) )
34 breq1 4450 . . . . . . . 8  |-  ( r  =  R  ->  (
r  Btwn  <. p ,  Q >.  <->  R  Btwn  <. p ,  Q >. ) )
3531, 33, 343orbi123d 1298 . . . . . . 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 710 . . . . . 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 2973 . . . . 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 2539 . . . . . . . 8  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
39383anbi1d 1303 . . . . . . 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 4450 . . . . . . . 8  |-  ( p  =  P  ->  (
p  Btwn  <. Q ,  R >. 
<->  P  Btwn  <. Q ,  R >. ) )
41 opeq2 4214 . . . . . . . . 9  |-  ( p  =  P  ->  <. R ,  p >.  =  <. R ,  P >. )
4241breq2d 4459 . . . . . . . 8  |-  ( p  =  P  ->  ( Q  Btwn  <. R ,  p >.  <-> 
Q  Btwn  <. R ,  P >. ) )
43 opeq1 4213 . . . . . . . . 9  |-  ( p  =  P  ->  <. p ,  Q >.  =  <. P ,  Q >. )
4443breq2d 4459 . . . . . . . 8  |-  ( p  =  P  ->  ( R  Btwn  <. p ,  Q >.  <-> 
R  Btwn  <. P ,  Q >. ) )
4540, 42, 443orbi123d 1298 . . . . . . 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 710 . . . . . 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 2973 . . . . 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 6375 . . . 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 253 . . 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 1198 . 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 354 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   <.cop 4033   class class class wbr 4447   `'ccnv 4998   ` cfv 5588   {coprab 6286   NNcn 10537   EEcee 23964    Btwn cbtwn 23965    Colinear ccolin 29540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-oprab 6289  df-colinear 29542
This theorem is referenced by:  brcolinear  29562
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