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

Theorem fvray 28184
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 6106 . 2  |-  ( PRay A )  =  (Ray
`  <. P ,  A >. )
2 eqid 2443 . . . . 5  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  =  {
x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }
3 fveq2 5703 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
43eleq2d 2510 . . . . . . . 8  |-  ( n  =  N  ->  ( P  e.  ( EE `  n )  <->  P  e.  ( EE `  N ) ) )
53eleq2d 2510 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
64, 53anbi12d 1290 . . . . . . 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 2978 . . . . . . . . 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 2454 . . . . . . 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 710 . . . . . 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 3085 . . . . 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 684 . . . 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 994 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  P  e.  ( EE `  N ) )
14 simpr2 995 . . . . 5  |-  ( ( N  e.  NN  /\  ( P  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  P  =/=  A ) )  ->  A  e.  ( EE `  N ) )
15 fvex 5713 . . . . . . 7  |-  ( EE
`  N )  e. 
_V
1615rabex 4455 . . . . . 6  |-  { x  e.  ( EE `  N
)  |  POutsideOf <. A ,  x >. }  e.  _V
17 eleq1 2503 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  e.  ( EE
`  n )  <->  P  e.  ( EE `  n ) ) )
18 neeq1 2628 . . . . . . . . . 10  |-  ( p  =  P  ->  (
p  =/=  a  <->  P  =/=  a ) )
1917, 183anbi13d 1291 . . . . . . . . 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 4307 . . . . . . . . . . 11  |-  ( p  =  P  ->  (
pOutsideOf <. a ,  x >.  <-> 
POutsideOf <. a ,  x >. ) )
2120rabbidv 2976 . . . . . . . . . 10  |-  ( p  =  P  ->  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. a ,  x >. } )
2221eqeq2d 2454 . . . . . . . . 9  |-  ( p  =  P  ->  (
r  =  { x  e.  ( EE `  n
)  |  pOutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. a ,  x >. } ) )
2319, 22anbi12d 710 . . . . . . . 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 2748 . . . . . . 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 2503 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
26 neeq2 2629 . . . . . . . . . 10  |-  ( a  =  A  ->  ( P  =/=  a  <->  P  =/=  A ) )
2725, 263anbi23d 1292 . . . . . . . . 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 4071 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  x >.  =  <. A ,  x >. )
2928breq2d 4316 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( POutsideOf
<. a ,  x >.  <->  POutsideOf <. A ,  x >. ) )
3029rabbidv 2976 . . . . . . . . . 10  |-  ( a  =  A  ->  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  =  {
x  e.  ( EE
`  n )  |  POutsideOf <. A ,  x >. } )
3130eqeq2d 2454 . . . . . . . . 9  |-  ( a  =  A  ->  (
r  =  { x  e.  ( EE `  n
)  |  POutsideOf <. a ,  x >. }  <->  r  =  { x  e.  ( EE `  n )  |  POutsideOf <. A ,  x >. } ) )
3227, 31anbi12d 710 . . . . . . . 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 2748 . . . . . . 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 2449 . . . . . . . . 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 703 . . . . . . . 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 2748 . . . . . . 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 6190 . . . . . 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 1303 . . . . 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 661 . . . 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 232 . . 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 4305 . . . . 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 28181 . . . . . 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 2507 . . . . 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 249 . . . 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 28183 . . . . 5  |-  Fun Ray
46 funbrfv 5742 . . . . 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 5 . . . 4  |-  ( <. P ,  A >.Ray { x  e.  ( EE
`  N )  |  POutsideOf <. A ,  x >. }  ->  (Ray `  <. P ,  A >. )  =  { x  e.  ( EE `  N )  |  POutsideOf <. A ,  x >. } )
4844, 47sylbir 213 . . 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 2487 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   E.wrex 2728   {crab 2731   _Vcvv 2984   <.cop 3895   class class class wbr 4304   Fun wfun 5424   ` cfv 5430  (class class class)co 6103   {coprab 6104   NNcn 10334   EEcee 23146  OutsideOfcoutsideof 28162  Raycray 28178
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 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-z 10659  df-uz 10874  df-fz 11450  df-ee 23149  df-ray 28181
This theorem is referenced by:  lineunray  28190
  Copyright terms: Public domain W3C validator