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Theorem colinearex 24683
Description: The colinear predicate exists. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
colinearex  |-  Colinear  e.  _V

Proof of Theorem colinearex
Dummy variables  a 
b  c  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-colinear 24664 . 2  |-  Colinear  =  `' { <. <. b ,  c
>. ,  a >.  |  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
>. ) ) }
2 nnex 9752 . . . . 5  |-  NN  e.  _V
3 fvex 5539 . . . . . . 7  |-  ( EE
`  n )  e. 
_V
43, 3xpex 4801 . . . . . 6  |-  ( ( EE `  n )  X.  ( EE `  n ) )  e. 
_V
54, 3xpex 4801 . . . . 5  |-  ( ( ( EE `  n
)  X.  ( EE
`  n ) )  X.  ( EE `  n ) )  e. 
_V
62, 5iunex 5770 . . . 4  |-  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) )  e.  _V
7 df-oprab 5862 . . . . 5  |-  { <. <.
b ,  c >. ,  a >.  |  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
>. ) ) }  =  { x  |  E. b E. c E. a
( x  =  <. <.
b ,  c >. ,  a >.  /\  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
>. ) ) ) }
8 opelxpi 4721 . . . . . . . . . . . . . 14  |-  ( ( b  e.  ( EE
`  n )  /\  c  e.  ( EE `  n ) )  ->  <. b ,  c >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )
983adant1 973 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  <. b ,  c >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )
10 simp1 955 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  a  e.  ( EE `  n
) )
11 opelxpi 4721 . . . . . . . . . . . . 13  |-  ( (
<. b ,  c >.  e.  ( ( EE `  n )  X.  ( EE `  n ) )  /\  a  e.  ( EE `  n ) )  ->  <. <. b ,  c >. ,  a
>.  e.  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
129, 10, 11syl2anc 642 . . . . . . . . . . . 12  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  <. <. b ,  c >. ,  a
>.  e.  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
1312adantr 451 . . . . . . . . . . 11  |-  ( ( ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n ) )  /\  ( a 
Btwn  <. b ,  c
>.  \/  b  Btwn  <. c ,  a >.  \/  c  Btwn  <. a ,  b
>. ) )  ->  <. <. b ,  c >. ,  a
>.  e.  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
1413reximi 2650 . . . . . . . . . 10  |-  ( 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
>. ) )  ->  E. n  e.  NN  <. <. b ,  c
>. ,  a >.  e.  ( ( ( EE
`  n )  X.  ( EE `  n
) )  X.  ( EE `  n ) ) )
15 eliun 3909 . . . . . . . . . 10  |-  ( <. <. b ,  c >. ,  a >.  e.  U_ n  e.  NN  (
( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n ) )  <->  E. n  e.  NN  <. <. b ,  c
>. ,  a >.  e.  ( ( ( EE
`  n )  X.  ( EE `  n
) )  X.  ( EE `  n ) ) )
1614, 15sylibr 203 . . . . . . . . 9  |-  ( 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
>. ) )  ->  <. <. b ,  c >. ,  a
>.  e.  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
17 eleq1 2343 . . . . . . . . . 10  |-  ( x  =  <. <. b ,  c
>. ,  a >.  -> 
( x  e.  U_ n  e.  NN  (
( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n ) )  <->  <. <. b ,  c >. ,  a
>.  e.  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) ) )
1817biimpar 471 . . . . . . . . 9  |-  ( ( x  =  <. <. b ,  c >. ,  a
>.  /\  <. <. b ,  c
>. ,  a >.  e. 
U_ n  e.  NN  ( ( ( EE
`  n )  X.  ( EE `  n
) )  X.  ( EE `  n ) ) )  ->  x  e.  U_ n  e.  NN  (
( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n ) ) )
1916, 18sylan2 460 . . . . . . . 8  |-  ( ( x  =  <. <. b ,  c >. ,  a
>.  /\  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
>. ) ) )  ->  x  e.  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
2019exlimiv 1666 . . . . . . 7  |-  ( E. a ( x  = 
<. <. b ,  c
>. ,  a >.  /\ 
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
>. ) ) )  ->  x  e.  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
2120exlimivv 1667 . . . . . 6  |-  ( E. b E. c E. a ( x  = 
<. <. b ,  c
>. ,  a >.  /\ 
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
>. ) ) )  ->  x  e.  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) ) )
2221abssi 3248 . . . . 5  |-  { x  |  E. b E. c E. a ( x  = 
<. <. b ,  c
>. ,  a >.  /\ 
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
>. ) ) ) } 
C_  U_ n  e.  NN  ( ( ( EE
`  n )  X.  ( EE `  n
) )  X.  ( EE `  n ) )
237, 22eqsstri 3208 . . . 4  |-  { <. <.
b ,  c >. ,  a >.  |  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
>. ) ) }  C_  U_ n  e.  NN  (
( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n ) )
246, 23ssexi 4159 . . 3  |-  { <. <.
b ,  c >. ,  a >.  |  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
>. ) ) }  e.  _V
2524cnvex 5209 . 2  |-  `' { <. <. b ,  c
>. ,  a >.  |  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
>. ) ) }  e.  _V
261, 25eqeltri 2353 1  |-  Colinear  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    \/ w3o 933    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   <.cop 3643   U_ciun 3905   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   ` cfv 5255   {coprab 5859   NNcn 9746   EEcee 24516    Btwn cbtwn 24517    Colinear ccolin 24660
This theorem is referenced by:  fvline  24767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-recs 6388  df-rdg 6423  df-nn 9747  df-colinear 24664
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