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Theorem colinearex 30898
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 30877 . 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 10637 . . . . 5  |-  NN  e.  _V
3 fvex 5889 . . . . . . 7  |-  ( EE
`  n )  e. 
_V
43, 3xpex 6614 . . . . . 6  |-  ( ( EE `  n )  X.  ( EE `  n ) )  e. 
_V
54, 3xpex 6614 . . . . 5  |-  ( ( ( EE `  n
)  X.  ( EE
`  n ) )  X.  ( EE `  n ) )  e. 
_V
62, 5iunex 6792 . . . 4  |-  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) )  e.  _V
7 df-oprab 6312 . . . . 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 4871 . . . . . . . . . . . . . 14  |-  ( ( b  e.  ( EE
`  n )  /\  c  e.  ( EE `  n ) )  ->  <. b ,  c >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )
983adant1 1048 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  <. b ,  c >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )
10 simp1 1030 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  a  e.  ( EE `  n
) )
11 opelxpi 4871 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 2852 . . . . . . . . . 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 4274 . . . . . . . . . 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 217 . . . . . . . . 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 2537 . . . . . . . . . 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 493 . . . . . . . . 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 482 . . . . . . . 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 1784 . . . . . . 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 1786 . . . . . 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 3490 . . . . 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 3448 . . . 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 4541 . . 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 6759 . 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 2545 1  |-  Colinear  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   E.wrex 2757   _Vcvv 3031   <.cop 3965   U_ciun 4269   class class class wbr 4395    X. cxp 4837   `'ccnv 4838   ` cfv 5589   {coprab 6309   NNcn 10631   EEcee 24997    Btwn cbtwn 24998    Colinear ccolin 30875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-i2m1 9625  ax-1ne0 9626  ax-rrecex 9629  ax-cnre 9630
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-nn 10632  df-colinear 30877
This theorem is referenced by:  fvline  30982
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