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Theorem linedegen 28319
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 6204 . 2  |-  ( ALine A )  =  (Line `  <. A ,  A >. )
2 neirr 2657 . . . . . . . . . . 11  |-  -.  A  =/=  A
3 simp3 990 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A )  ->  A  =/=  A )
42, 3mto 176 . . . . . . . . . 10  |-  -.  ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )
54intnanr 906 . . . . . . . . 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 2924 . . . . . . 7  |-  -.  E. n  e.  NN  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
87nex 1601 . . . . . 6  |-  -.  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
9 eleq1 2526 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
10 neeq1 2733 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
119, 103anbi13d 1292 . . . . . . . . . . 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 4168 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
13 eceq1 7248 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  =  <. A ,  y
>.  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1412, 13syl 16 . . . . . . . . . . . 12  |-  ( x  =  A  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1514eqeq2d 2468 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
l  =  [ <. x ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  y
>. ] `'  Colinear  ) )
1611, 15anbi12d 710 . . . . . . . . . 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  ) ) )
1716rexbidv 2868 . . . . . . . . 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  ) ) )
1817exbidv 1681 . . . . . . . 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  ) ) )
19 eleq1 2526 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
20 neeq2 2735 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( A  =/=  y  <->  A  =/=  A ) )
2119, 203anbi23d 1293 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( A  e.  ( EE `  n )  /\  y  e.  ( EE `  n )  /\  A  =/=  y
)  <->  ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
) ) )
22 opeq2 4169 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  <. A , 
y >.  =  <. A ,  A >. )
23 eceq1 7248 . . . . . . . . . . . . 13  |-  ( <. A ,  y >.  = 
<. A ,  A >.  ->  [ <. A ,  y
>. ] `'  Colinear  =  [ <. A ,  A >. ] `' 
Colinear  )
2422, 23syl 16 . . . . . . . . . . . 12  |-  ( y  =  A  ->  [ <. A ,  y >. ] `'  Colinear  =  [ <. A ,  A >. ] `'  Colinear  )
2524eqeq2d 2468 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
l  =  [ <. A ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  A >. ] `'  Colinear  ) )
2621, 25anbi12d 710 . . . . . . . . . 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  ) ) )
2726rexbidv 2868 . . . . . . . . 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  ) ) )
2827exbidv 1681 . . . . . . . 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  ) ) )
2918, 28opelopabg 4716 . . . . . . 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  ) ) )
3029anidms 645 . . . . . 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  ) ) )
318, 30mtbiri 303 . . . . 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  ) } )
32 relopab 5075 . . . . . . . . 9  |-  Rel  { <. x ,  y >.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n
)  /\  x  =/=  y )  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
33 df-rel 4956 . . . . . . . . 9  |-  ( Rel 
{ <. x ,  y
>.  |  E. 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  ) } 
C_  ( _V  X.  _V ) )
3432, 33mpbi 208 . . . . . . . 8  |-  { <. x ,  y >.  |  E. l E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) } 
C_  ( _V  X.  _V )
3534sseli 3461 . . . . . . 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 ) )
36 opelxp1 4981 . . . . . . 7  |-  ( <. A ,  A >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
3735, 36syl 16 . . . . . 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 )
3837con3i 135 . . . . 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  ) } )
3931, 38pm2.61i 164 . . . 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  ) }
40 df-line2 28313 . . . . . . 7  |- Line  =  { <. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
4140dmeqi 5150 . . . . . 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  ) }
42 dmoprab 6282 . . . . . 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  ) }
4341, 42eqtri 2483 . . . . 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  ) }
4443eleq2i 2532 . . . 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  ) } )
4539, 44mtbir 299 . . 3  |-  -.  <. A ,  A >.  e.  dom Line
46 ndmfv 5824 . . 3  |-  ( -. 
<. A ,  A >.  e. 
dom Line  ->  (Line `  <. A ,  A >. )  =  (/) )
4745, 46ax-mp 5 . 2  |-  (Line `  <. A ,  A >. )  =  (/)
481, 47eqtri 2483 1  |-  ( ALine A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   E.wrex 2800   _Vcvv 3078    C_ wss 3437   (/)c0 3746   <.cop 3992   {copab 4458    X. cxp 4947   `'ccnv 4948   dom cdm 4949   Rel wrel 4954   ` cfv 5527  (class class class)co 6201   {coprab 6202   [cec 7210   NNcn 10434   EEcee 23287    Colinear ccolin 28213  Linecline2 28310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fv 5535  df-ov 6204  df-oprab 6205  df-ec 7214  df-line2 28313
This theorem is referenced by: (None)
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