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Theorem colinearex 25994
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 25975 . 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 10006 . . . . 5  |-  NN  e.  _V
3 fvex 5742 . . . . . . 7  |-  ( EE
`  n )  e. 
_V
43, 3xpex 4990 . . . . . 6  |-  ( ( EE `  n )  X.  ( EE `  n ) )  e. 
_V
54, 3xpex 4990 . . . . 5  |-  ( ( ( EE `  n
)  X.  ( EE
`  n ) )  X.  ( EE `  n ) )  e. 
_V
62, 5iunex 5991 . . . 4  |-  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) )  e.  _V
7 df-oprab 6085 . . . . 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 4910 . . . . . . . . . . . . . 14  |-  ( ( b  e.  ( EE
`  n )  /\  c  e.  ( EE `  n ) )  ->  <. b ,  c >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )
983adant1 975 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  <. b ,  c >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )
10 simp1 957 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  a  e.  ( EE `  n
) )
11 opelxpi 4910 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 2813 . . . . . . . . . 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 4097 . . . . . . . . . 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 204 . . . . . . . . 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 2496 . . . . . . . . . 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 472 . . . . . . . . 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 461 . . . . . . . 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 1644 . . . . . . 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 1645 . . . . . 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 3418 . . . . 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 3378 . . . 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 4348 . . 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 5406 . 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 2506 1  |-  Colinear  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 359    \/ w3o 935    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956   <.cop 3817   U_ciun 4093   class class class wbr 4212    X. cxp 4876   `'ccnv 4877   ` cfv 5454   {coprab 6082   NNcn 10000   EEcee 25827    Btwn cbtwn 25828    Colinear ccolin 25971
This theorem is referenced by:  fvline  26078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-recs 6633  df-rdg 6668  df-nn 10001  df-colinear 25975
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