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Theorem fvray 25782
Description: Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvray  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
( PRay A )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } )
Distinct variable groups:    x, A    x, N    x, P

Proof of Theorem fvray
Dummy variables  a  n  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 6016 . 2  |-  ( PRay A )  =  (Ray
`  <. P ,  A >. )
2 eqid 2380 . . . . 5  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }
3 fveq2 5661 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
43eleq2d 2447 . . . . . . . 8  |-  ( n  =  N  ->  ( P  e.  ( EE `  n )  <->  P  e.  ( EE `  N ) ) )
53eleq2d 2447 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
64, 53anbi12d 1255 . . . . . . 7  |-  ( n  =  N  ->  (
( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  <->  ( P  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  P  =/=  A
) ) )
7 rabeq 2886 . . . . . . . . 9  |-  ( ( EE `  n )  =  ( EE `  N )  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
83, 7syl 16 . . . . . . . 8  |-  ( n  =  N  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
98eqeq2d 2391 . . . . . . 7  |-  ( n  =  N  ->  ( { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. }  =  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  <->  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } ) )
106, 9anbi12d 692 . . . . . 6  |-  ( n  =  N  ->  (
( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )  <->  ( ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } ) ) )
1110rspcev 2988 . . . . 5  |-  ( ( N  e.  NN  /\  ( ( P  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } ) )  ->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) )
122, 11mpanr2 666 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) )
13 simpr1 963 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  P  e.  ( EE `  N ) )
14 simpr2 964 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  A  e.  ( EE `  N ) )
15 fvex 5675 . . . . . . 7  |-  ( EE
`  N )  e. 
_V
1615rabex 4288 . . . . . 6  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  e.  _V
17 eleq1 2440 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
18 neeq1 2551 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  =/=  a  <->  P  =/=  a ) )
1917, 183anbi13d 1256 . . . . . . . . 9  |-  ( p  =  P  ->  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  <->  ( P  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  P  =/=  a
) ) )
20 breq1 4149 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
pOutsideOf <. a ,  x >.  <-> 
POutsideOf <. a ,  x >. ) )
2120rabbidv 2884 . . . . . . . . . 10  |-  ( p  =  P  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. a ,  x >. } )
2221eqeq2d 2391 . . . . . . . . 9  |-  ( p  =  P  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) )
2319, 22anbi12d 692 . . . . . . . 8  |-  ( p  =  P  ->  (
( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  ( ( P  e.  ( EE `  n )  /\  a  e.  ( EE `  n
)  /\  P  =/=  a )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) ) )
2423rexbidv 2663 . . . . . . 7  |-  ( p  =  P  ->  ( E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  a  e.  ( EE `  n
)  /\  P  =/=  a )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) ) )
25 eleq1 2440 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
26 neeq2 2552 . . . . . . . . . 10  |-  ( a  =  A  ->  ( P  =/=  a  <->  P  =/=  A ) )
2725, 263anbi23d 1257 . . . . . . . . 9  |-  ( a  =  A  ->  (
( P  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  P  =/=  a
)  <->  ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
) ) )
28 opeq1 3919 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  x >.  =  <. A ,  x >. )
2928breq2d 4158 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( POutsideOf
<. a ,  x >.  <->  POutsideOf <. A ,  x >. ) )
3029rabbidv 2884 . . . . . . . . . 10  |-  ( a  =  A  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )
3130eqeq2d 2391 . . . . . . . . 9  |-  ( a  =  A  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) )
3227, 31anbi12d 692 . . . . . . . 8  |-  ( a  =  A  ->  (
( ( P  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  P  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } )  <->  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) ) )
3332rexbidv 2663 . . . . . . 7  |-  ( a  =  A  ->  ( E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  P  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) ) )
34 eqeq1 2386 . . . . . . . . 9  |-  ( r  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. A ,  x >. }  <->  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) )
3534anbi2d 685 . . . . . . . 8  |-  ( r  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  ->  (
( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } )  <->  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3635rexbidv 2663 . . . . . . 7  |-  ( r  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  ->  ( E. n  e.  NN  ( ( P  e.  ( EE `  n
)  /\  A  e.  ( EE `  n )  /\  P  =/=  A
)  /\  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } )  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3724, 33, 36eloprabg 6093 . . . . . 6  |-  ( ( P  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  e.  _V )  ->  ( <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3816, 37mp3an3 1268 . . . . 5  |-  ( ( P  e.  ( EE
`  N )  /\  A  e.  ( EE `  N ) )  -> 
( <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
3913, 14, 38syl2anc 643 . . . 4  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
( <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  <->  E. n  e.  NN  ( ( P  e.  ( EE `  n )  /\  A  e.  ( EE `  n
)  /\  P  =/=  A )  /\  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } ) ) )
4012, 39mpbird 224 . . 3  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) } )
41 df-br 4147 . . . . 5  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  <->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e. Ray )
42 df-ray 25779 . . . . . 6  |- Ray  =  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }
4342eleq2i 2444 . . . . 5  |-  ( <. <. P ,  A >. ,  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } >.  e. Ray  <->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) } )
4441, 43bitri 241 . . . 4  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  <->  <. <. P ,  A >. ,  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } >.  e.  { <. <. p ,  a
>. ,  r >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) } )
45 funray 25781 . . . . 5  |-  Fun Ray
46 funbrfv 5697 . . . . 5  |-  ( Fun Ray  ->  ( <. P ,  A >.Ray { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. }  ->  (Ray `  <. P ,  A >. )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } ) )
4745, 46ax-mp 8 . . . 4  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  ->  (Ray `  <. P ,  A >. )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } )
4844, 47sylbir 205 . . 3  |-  ( <. <. P ,  A >. ,  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } >.  e.  { <. <.
p ,  a >. ,  r >.  |  E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  p  =/=  a
)  /\  r  =  { x  e.  ( EE `  n )  |  pOutsideOf <. a ,  x >. } ) }  ->  (Ray
`  <. P ,  A >. )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } )
4940, 48syl 16 . 2  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
(Ray `  <. P ,  A >. )  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. } )
501, 49syl5eq 2424 1  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  -> 
( PRay A )  =  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   {crab 2646   _Vcvv 2892   <.cop 3753   class class class wbr 4146   Fun wfun 5381   ` cfv 5387  (class class class)co 6013   {coprab 6014   NNcn 9925   EEcee 25534  OutsideOfcoutsideof 25760  Raycray 25776
This theorem is referenced by:  lineunray  25788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-z 10208  df-uz 10414  df-fz 10969  df-ee 25537  df-ray 25779
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