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Theorem colinearex 29315
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 29294 . 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 10542 . . . . 5  |-  NN  e.  _V
3 fvex 5876 . . . . . . 7  |-  ( EE
`  n )  e. 
_V
43, 3xpex 6588 . . . . . 6  |-  ( ( EE `  n )  X.  ( EE `  n ) )  e. 
_V
54, 3xpex 6588 . . . . 5  |-  ( ( ( EE `  n
)  X.  ( EE
`  n ) )  X.  ( EE `  n ) )  e. 
_V
62, 5iunex 6764 . . . 4  |-  U_ n  e.  NN  ( ( ( EE `  n )  X.  ( EE `  n ) )  X.  ( EE `  n
) )  e.  _V
7 df-oprab 6288 . . . . 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 5031 . . . . . . . . . . . . . 14  |-  ( ( b  e.  ( EE
`  n )  /\  c  e.  ( EE `  n ) )  ->  <. b ,  c >.  e.  ( ( EE `  n )  X.  ( EE `  n ) ) )
983adant1 1014 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  <. b ,  c >.  e.  ( ( EE `  n
)  X.  ( EE
`  n ) ) )
10 simp1 996 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  ->  a  e.  ( EE `  n
) )
11 opelxpi 5031 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 2932 . . . . . . . . . 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 4330 . . . . . . . . . 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 212 . . . . . . . . 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 2539 . . . . . . . . . 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 485 . . . . . . . . 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 474 . . . . . . . 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 1698 . . . . . . 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 1699 . . . . . 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 3575 . . . . 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 3534 . . . 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 4592 . . 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 6731 . 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 2551 1  |-  Colinear  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   E.wrex 2815   _Vcvv 3113   <.cop 4033   U_ciun 4325   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   ` cfv 5588   {coprab 6285   NNcn 10536   EEcee 23895    Btwn cbtwn 23896    Colinear ccolin 29292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rrecex 9564  ax-cnre 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-om 6685  df-recs 7042  df-rdg 7076  df-nn 10537  df-colinear 29294
This theorem is referenced by:  fvline  29399
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