Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  linedegen Structured version   Unicode version

Theorem linedegen 28125
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 6089 . 2  |-  ( ALine A )  =  (Line `  <. A ,  A >. )
2 neirr 2608 . . . . . . . . . . 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 2813 . . . . . . 7  |-  -.  E. n  e.  NN  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
87nex 1600 . . . . . 6  |-  -.  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
9 eleq1 2498 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
10 neeq1 2611 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
119, 103anbi13d 1291 . . . . . . . . . . 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 4054 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
13 eceq1 7129 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  =  <. A ,  y
>.  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1412, 13syl 16 . . . . . . . . . . . 12  |-  ( x  =  A  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1514eqeq2d 2449 . . . . . . . . . . 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 2731 . . . . . . . . 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 1680 . . . . . . . 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 2498 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
20 neeq2 2612 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( A  =/=  y  <->  A  =/=  A ) )
2119, 203anbi23d 1292 . . . . . . . . . . 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 4055 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  <. A , 
y >.  =  <. A ,  A >. )
23 eceq1 7129 . . . . . . . . . . . . 13  |-  ( <. A ,  y >.  = 
<. A ,  A >.  ->  [ <. A ,  y
>. ] `'  Colinear  =  [ <. A ,  A >. ] `' 
Colinear  )
2422, 23syl 16 . . . . . . . . . . . 12  |-  ( y  =  A  ->  [ <. A ,  y >. ] `'  Colinear  =  [ <. A ,  A >. ] `'  Colinear  )
2524eqeq2d 2449 . . . . . . . . . . 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 2731 . . . . . . . . 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 1680 . . . . . . . 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 4602 . . . . . . 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 4961 . . . . . . . . 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 4842 . . . . . . . . 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 3347 . . . . . . 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 4867 . . . . . . 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 28119 . . . . . . 7  |- Line  =  { <. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
4140dmeqi 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  ) }
42 dmoprab 6166 . . . . . 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 2458 . . . . 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 2502 . . . 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 5709 . . 3  |-  ( -. 
<. A ,  A >.  e. 
dom Line  ->  (Line `  <. A ,  A >. )  =  (/) )
4745, 46ax-mp 5 . 2  |-  (Line `  <. A ,  A >. )  =  (/)
481, 47eqtri 2458 1  |-  ( ALine A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   E.wrex 2711   _Vcvv 2967    C_ wss 3323   (/)c0 3632   <.cop 3878   {copab 4344    X. cxp 4833   `'ccnv 4834   dom cdm 4835   Rel wrel 4840   ` cfv 5413  (class class class)co 6086   {coprab 6087   [cec 7091   NNcn 10314   EEcee 23085    Colinear ccolin 28019  Linecline2 28116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fv 5421  df-ov 6089  df-oprab 6090  df-ec 7095  df-line2 28119
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator