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Theorem colinearex 24755
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 24736 . 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 9768 . . . . 5  |-  NN  e.  _V
3 fvex 5555 . . . . . . 7  |-  ( EE
`  n )  e. 
_V
43, 3xpex 4817 . . . . . 6  |-  ( ( EE `  n )  X.  ( EE `  n ) )  e. 
_V
54, 3xpex 4817 . . . . 5  |-  ( ( ( EE `  n
)  X.  ( EE
`  n ) )  X.  ( EE `  n ) )  e. 
_V
62, 5iunex 5786 . . . 4  |-  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) )  e.  _V
7 df-oprab 5878 . . . . 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 4737 . . . . . . . . . . . . . 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 4737 . . . . . . . . . . . . 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 2663 . . . . . . . . . 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 3925 . . . . . . . . . 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 2356 . . . . . . . . . 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 1624 . . . . . . 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 1625 . . . . . 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 3261 . . . . 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 3221 . . . 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 4175 . . 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 5225 . 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 2366 1  |-  Colinear  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    \/ w3o 933    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   <.cop 3656   U_ciun 3921   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   ` cfv 5271   {coprab 5875   NNcn 9762   EEcee 24588    Btwn cbtwn 24589    Colinear ccolin 24732
This theorem is referenced by:  fvline  24839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-recs 6404  df-rdg 6439  df-nn 9763  df-colinear 24736
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