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Theorem linedegen 25152
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 5903 . 2  |-  ( ALine A )  =  (Line `  <. A ,  A >. )
2 neirr 2484 . . . . . . . . . . 11  |-  -.  A  =/=  A
3 simp3 957 . . . . . . . . . . 11  |-  ( ( A  e.  ( EE
`  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A )  ->  A  =/=  A )
42, 3mto 167 . . . . . . . . . 10  |-  -.  ( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  A  =/=  A )
54intnanr 881 . . . . . . . . 9  |-  -.  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
65a1i 10 . . . . . . . 8  |-  ( n  e.  NN  ->  -.  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  ) )
76nrex 2679 . . . . . . 7  |-  -.  E. n  e.  NN  (
( A  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
87nex 1546 . . . . . 6  |-  -.  E. l E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  A  =/=  A
)  /\  l  =  [ <. A ,  A >. ] `'  Colinear  )
9 eleq1 2376 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
10 neeq1 2487 . . . . . . . . . . . 12  |-  ( x  =  A  ->  (
x  =/=  y  <->  A  =/=  y ) )
119, 103anbi13d 1254 . . . . . . . . . . 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 3833 . . . . . . . . . . . . 13  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
13 eceq1 6738 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  =  <. A ,  y
>.  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1412, 13syl 15 . . . . . . . . . . . 12  |-  ( x  =  A  ->  [ <. x ,  y >. ] `'  Colinear  =  [ <. A ,  y
>. ] `'  Colinear  )
1514eqeq2d 2327 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
l  =  [ <. x ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  y
>. ] `'  Colinear  ) )
1611, 15anbi12d 691 . . . . . . . . . 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 2598 . . . . . . . . 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 1617 . . . . . . . 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 2376 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
20 neeq2 2488 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( A  =/=  y  <->  A  =/=  A ) )
2119, 203anbi23d 1255 . . . . . . . . . . 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 3834 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  <. A , 
y >.  =  <. A ,  A >. )
23 eceq1 6738 . . . . . . . . . . . . 13  |-  ( <. A ,  y >.  = 
<. A ,  A >.  ->  [ <. A ,  y
>. ] `'  Colinear  =  [ <. A ,  A >. ] `' 
Colinear  )
2422, 23syl 15 . . . . . . . . . . . 12  |-  ( y  =  A  ->  [ <. A ,  y >. ] `'  Colinear  =  [ <. A ,  A >. ] `'  Colinear  )
2524eqeq2d 2327 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
l  =  [ <. A ,  y >. ] `'  Colinear  <->  l  =  [ <. A ,  A >. ] `'  Colinear  ) )
2621, 25anbi12d 691 . . . . . . . . . 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 2598 . . . . . . . . 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 1617 . . . . . . . 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 4320 . . . . . . 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 626 . . . . . 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 294 . . . . 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 4849 . . . . . . . . 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 4733 . . . . . . . . 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 199 . . . . . . . 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 3210 . . . . . . 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 4759 . . . . . . 7  |-  ( <. A ,  A >.  e.  ( _V  X.  _V )  ->  A  e.  _V )
3735, 36syl 15 . . . . . 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 127 . . . . 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 156 . . . 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 25146 . . . . . . 7  |- Line  =  { <. <. x ,  y
>. ,  l >.  |  E. n  e.  NN  ( ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n )  /\  x  =/=  y
)  /\  l  =  [ <. x ,  y
>. ] `'  Colinear  ) }
4140dmeqi 4917 . . . . . 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 5970 . . . . . 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 2336 . . . . 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 2380 . . . 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 290 . . 3  |-  -.  <. A ,  A >.  e.  dom Line
46 ndmfv 5590 . . 3  |-  ( -. 
<. A ,  A >.  e. 
dom Line  ->  (Line `  <. A ,  A >. )  =  (/) )
4745, 46ax-mp 8 . 2  |-  (Line `  <. A ,  A >. )  =  (/)
481, 47eqtri 2336 1  |-  ( ALine A )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   _Vcvv 2822    C_ wss 3186   (/)c0 3489   <.cop 3677   {copab 4113    X. cxp 4724   `'ccnv 4725   dom cdm 4726   Rel wrel 4731   ` cfv 5292  (class class class)co 5900   {coprab 5901   [cec 6700   NNcn 9791   EEcee 24902    Colinear ccolin 25046  Linecline2 25143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fv 5300  df-ov 5903  df-oprab 5904  df-ec 6704  df-line2 25146
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