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Theorem linedegen 30910
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 6293 . 2  |-  ( ALine A )  =  (Line `  <. A ,  A >. )
2 neirr 2632 . . . . . . . . . . 11  |-  -.  A  =/=  A
3 simp3 1010 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A )  ->  A  =/=  A )
42, 3mto 180 . . . . . . . . . 10  |-  -.  ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )
54intnanr 926 . . . . . . . . 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 2842 . . . . . . 7  |-  -.  E. n  e.  NN  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
87nex 1678 . . . . . 6  |-  -.  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
9 eleq1 2517 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
10 neeq1 2686 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
119, 103anbi13d 1341 . . . . . . . . . . 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 4166 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
1312eceq1d 7400 . . . . . . . . . . . 12  |-  ( x  =  A  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1413eqeq2d 2461 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
l  =  [ <. x ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  y
>. ] `'  Colinear  ) )
1511, 14anbi12d 717 . . . . . . . . . 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 2901 . . . . . . . . 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 1768 . . . . . . . 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 2517 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
19 neeq2 2687 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( A  =/=  y  <->  A  =/=  A ) )
2018, 193anbi23d 1342 . . . . . . . . . . 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 4167 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  <. A , 
y >.  =  <. A ,  A >. )
2221eceq1d 7400 . . . . . . . . . . . 12  |-  ( y  =  A  ->  [ <. A ,  y >. ] `'  Colinear  =  [ <. A ,  A >. ] `'  Colinear  )
2322eqeq2d 2461 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
l  =  [ <. A ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  A >. ] `'  Colinear  ) )
2420, 23anbi12d 717 . . . . . . . . . 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 2901 . . . . . . . . 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 1768 . . . . . . . 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 651 . . . . . 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 305 . . . . 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 4907 . . . . . . 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 4867 . . . . . . 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 141 . . . . 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 168 . . . 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 30904 . . . . . . 7  |- Line  =  { <. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
3635dmeqi 5036 . . . . . 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 6377 . . . . . 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 2473 . . . . 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 2521 . . . 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 301 . . 3  |-  -.  <. A ,  A >.  e.  dom Line
41 ndmfv 5889 . . 3  |-  ( -. 
<. A ,  A >.  e. 
dom Line  ->  (Line `  <. A ,  A >. )  =  (/) )
4240, 41ax-mp 5 . 2  |-  (Line `  <. A ,  A >. )  =  (/)
431, 42eqtri 2473 1  |-  ( ALine A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045   (/)c0 3731   <.cop 3974   {copab 4460    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ` cfv 5582  (class class class)co 6290   {coprab 6291   [cec 7361   NNcn 10609   EEcee 24918    Colinear ccolin 30804  Linecline2 30901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fv 5590  df-ov 6293  df-oprab 6294  df-ec 7365  df-line2 30904
This theorem is referenced by: (None)
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