Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcolinear2 Structured version   Unicode version

Theorem brcolinear2 30636
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 30635 . . . 4  |-  Rel  Colinear
21brrelexi 4886 . . 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 3087 . . . . . 6  |-  ( P  e.  ( EE `  n )  ->  P  e.  _V )
543ad2ant1 1026 . . . . 5  |-  ( ( P  e.  ( EE
`  n )  /\  Q  e.  ( EE `  n )  /\  R  e.  ( EE `  n
) )  ->  P  e.  _V )
65adantr 466 . . . 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 2912 . . 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 4418 . . . . . 6  |-  ( P 
Colinear 
<. Q ,  R >.  <->  <. P ,  <. Q ,  R >. >.  e.  Colinear  )
10 df-colinear 30617 . . . . . . 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 2498 . . . . . 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 252 . . . . 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 4677 . . . . . . 7  |-  <. Q ,  R >.  e.  _V
14 opelcnvg 5025 . . . . . . 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 675 . . . . . 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 1028 . . . . 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 260 . . . 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 2492 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  e.  ( EE
`  n )  <->  Q  e.  ( EE `  n ) ) )
19183anbi2d 1340 . . . . . . 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 4181 . . . . . . . . 9  |-  ( q  =  Q  ->  <. q ,  r >.  =  <. Q ,  r >. )
2120breq2d 4429 . . . . . . . 8  |-  ( q  =  Q  ->  (
p  Btwn  <. q ,  r >.  <->  p  Btwn  <. Q , 
r >. ) )
22 breq1 4420 . . . . . . . 8  |-  ( q  =  Q  ->  (
q  Btwn  <. r ,  p >.  <->  Q  Btwn  <. r ,  p >. ) )
23 opeq2 4182 . . . . . . . . 9  |-  ( q  =  Q  ->  <. p ,  q >.  =  <. p ,  Q >. )
2423breq2d 4429 . . . . . . . 8  |-  ( q  =  Q  ->  (
r  Btwn  <. p ,  q >.  <->  r  Btwn  <. p ,  Q >. ) )
2521, 22, 243orbi123d 1334 . . . . . . 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 715 . . . . . 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 2937 . . . . 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 2492 . . . . . . . 8  |-  ( r  =  R  ->  (
r  e.  ( EE
`  n )  <->  R  e.  ( EE `  n ) ) )
29283anbi3d 1341 . . . . . . 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 4182 . . . . . . . . 9  |-  ( r  =  R  ->  <. Q , 
r >.  =  <. Q ,  R >. )
3130breq2d 4429 . . . . . . . 8  |-  ( r  =  R  ->  (
p  Btwn  <. Q , 
r >. 
<->  p  Btwn  <. Q ,  R >. ) )
32 opeq1 4181 . . . . . . . . 9  |-  ( r  =  R  ->  <. r ,  p >.  =  <. R ,  p >. )
3332breq2d 4429 . . . . . . . 8  |-  ( r  =  R  ->  ( Q  Btwn  <. r ,  p >.  <-> 
Q  Btwn  <. R ,  p >. ) )
34 breq1 4420 . . . . . . . 8  |-  ( r  =  R  ->  (
r  Btwn  <. p ,  Q >.  <->  R  Btwn  <. p ,  Q >. ) )
3531, 33, 343orbi123d 1334 . . . . . . 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 715 . . . . . 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 2937 . . . . 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 2492 . . . . . . . 8  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
39383anbi1d 1339 . . . . . . 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 4420 . . . . . . . 8  |-  ( p  =  P  ->  (
p  Btwn  <. Q ,  R >. 
<->  P  Btwn  <. Q ,  R >. ) )
41 opeq2 4182 . . . . . . . . 9  |-  ( p  =  P  ->  <. R ,  p >.  =  <. R ,  P >. )
4241breq2d 4429 . . . . . . . 8  |-  ( p  =  P  ->  ( Q  Btwn  <. R ,  p >.  <-> 
Q  Btwn  <. R ,  P >. ) )
43 opeq1 4181 . . . . . . . . 9  |-  ( p  =  P  ->  <. p ,  Q >.  =  <. P ,  Q >. )
4443breq2d 4429 . . . . . . . 8  |-  ( p  =  P  ->  ( R  Btwn  <. p ,  Q >.  <-> 
R  Btwn  <. P ,  Q >. ) )
4540, 42, 443orbi123d 1334 . . . . . . 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 715 . . . . . 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 2937 . . . . 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 6389 . . . 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 256 . . 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 1207 . 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 355 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 187    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774   _Vcvv 3078   <.cop 3999   class class class wbr 4417   `'ccnv 4844   ` cfv 5592   {coprab 6297   NNcn 10598   EEcee 24790    Btwn cbtwn 24791    Colinear ccolin 30615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-rel 4852  df-cnv 4853  df-oprab 6300  df-colinear 30617
This theorem is referenced by:  brcolinear  30637
  Copyright terms: Public domain W3C validator