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Theorem linedegen 30981
Description: When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
linedegen  |-  ( ALine A )  =  (/)

Proof of Theorem linedegen
Dummy variables  l  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6311 . 2  |-  ( ALine A )  =  (Line `  <. A ,  A >. )
2 neirr 2652 . . . . . . . . . . 11  |-  -.  A  =/=  A
3 simp3 1032 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A )  ->  A  =/=  A )
42, 3mto 181 . . . . . . . . . 10  |-  -.  ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )
54intnanr 929 . . . . . . . . 9  |-  -.  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
65a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  -.  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) )
76nrex 2841 . . . . . . 7  |-  -.  E. n  e.  NN  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
87nex 1686 . . . . . 6  |-  -.  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
9 eleq1 2537 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
10 neeq1 2705 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
119, 103anbi13d 1367 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  <->  ( A  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  A  =/=  y
) ) )
12 opeq1 4158 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
1312eceq1d 7418 . . . . . . . . . . . 12  |-  ( x  =  A  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1413eqeq2d 2481 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
l  =  [ <. x ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  y
>. ] `'  Colinear  ) )
1511, 14anbi12d 725 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  )  <->  ( ( A  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  A  =/=  y )  /\  l  =  [ <. A ,  y
>. ] `'  Colinear  ) ) )
1615rexbidv 2892 . . . . . . . . 9  |-  ( x  =  A  ->  ( E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  )  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  A  =/=  y )  /\  l  =  [ <. A ,  y
>. ] `'  Colinear  ) ) )
1716exbidv 1776 . . . . . . . 8  |-  ( x  =  A  ->  ( E. l E. n  e.  NN  ( ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  x  =/=  y )  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  )  <->  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  A  =/=  y
)  /\  l  =  [ <. A ,  y
>. ] `'  Colinear  ) ) )
18 eleq1 2537 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
19 neeq2 2706 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( A  =/=  y  <->  A  =/=  A ) )
2018, 193anbi23d 1368 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( A  e.  ( EE `  n )  /\  y  e.  ( EE `  n )  /\  A  =/=  y
)  <->  ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
) ) )
21 opeq2 4159 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  <. A , 
y >.  =  <. A ,  A >. )
2221eceq1d 7418 . . . . . . . . . . . 12  |-  ( y  =  A  ->  [ <. A ,  y >. ] `'  Colinear  =  [ <. A ,  A >. ] `'  Colinear  )
2322eqeq2d 2481 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
l  =  [ <. A ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  A >. ] `'  Colinear  ) )
2420, 23anbi12d 725 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( ( A  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  A  =/=  y
)  /\  l  =  [ <. A ,  y
>. ] `'  Colinear  )  <->  ( ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) ) )
2524rexbidv 2892 . . . . . . . . 9  |-  ( y  =  A  ->  ( E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  A  =/=  y
)  /\  l  =  [ <. A ,  y
>. ] `'  Colinear  )  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) ) )
2625exbidv 1776 . . . . . . . 8  |-  ( y  =  A  ->  ( E. l E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  A  =/=  y )  /\  l  =  [ <. A ,  y
>. ] `'  Colinear  )  <->  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) ) )
2717, 26opelopabg 4719 . . . . . . 7  |-  ( ( A  e.  _V  /\  A  e.  _V )  ->  ( <. A ,  A >.  e.  { <. x ,  y >.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }  <->  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) ) )
2827anidms 657 . . . . . 6  |-  ( A  e.  _V  ->  ( <. A ,  A >.  e. 
{ <. x ,  y
>.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }  <->  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) ) )
298, 28mtbiri 310 . . . . 5  |-  ( A  e.  _V  ->  -.  <. A ,  A >.  e. 
{ <. x ,  y
>.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) } )
30 elopaelxp 4912 . . . . . . 7  |-  ( <. A ,  A >.  e. 
{ <. x ,  y
>.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }  ->  <. A ,  A >.  e.  ( _V  X.  _V ) )
31 opelxp1 4872 . . . . . . 7  |-  ( <. A ,  A >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
3230, 31syl 17 . . . . . 6  |-  ( <. A ,  A >.  e. 
{ <. x ,  y
>.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }  ->  A  e.  _V )
3332con3i 142 . . . . 5  |-  ( -.  A  e.  _V  ->  -. 
<. A ,  A >.  e. 
{ <. x ,  y
>.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) } )
3429, 33pm2.61i 169 . . . 4  |-  -.  <. A ,  A >.  e.  { <. x ,  y >.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  x  =/=  y )  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
35 df-line2 30975 . . . . . . 7  |- Line  =  { <. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
3635dmeqi 5041 . . . . . 6  |-  dom Line  =  dom  {
<. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
37 dmoprab 6396 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }  =  { <. x ,  y >.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
3836, 37eqtri 2493 . . . . 5  |-  dom Line  =  { <. x ,  y >.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  x  =/=  y )  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
3938eleq2i 2541 . . . 4  |-  ( <. A ,  A >.  e. 
dom Line 
<-> 
<. A ,  A >.  e. 
{ <. x ,  y
>.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) } )
4034, 39mtbir 306 . . 3  |-  -.  <. A ,  A >.  e.  dom Line
41 ndmfv 5903 . . 3  |-  ( -. 
<. A ,  A >.  e. 
dom Line  ->  (Line `  <. A ,  A >. )  =  (/) )
4240, 41ax-mp 5 . 2  |-  (Line `  <. A ,  A >. )  =  (/)
431, 42eqtri 2493 1  |-  ( ALine A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031   (/)c0 3722   <.cop 3965   {copab 4453    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ` cfv 5589  (class class class)co 6308   {coprab 6309   [cec 7379   NNcn 10631   EEcee 24997    Colinear ccolin 30875  Linecline2 30972
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-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fv 5597  df-ov 6311  df-oprab 6312  df-ec 7383  df-line2 30975
This theorem is referenced by: (None)
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