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Theorem isray 26154
Description: A non-degenerated ray. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isray.1  |-  P  =  (PPoints `  G )
isray.2  |-  S  =  ( seg `  G
)
isray.3  |-  R  =  (ray `  G )
isray.5  |-  ( ph  ->  G  e. Ibg )
isray.6a  |-  ( ph  ->  X  e.  P )
isray.6b  |-  ( ph  ->  Y  e.  P )
isray.4  |-  B  =  (btw `  G )
isray.7  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
isray  |-  ( ph  ->  ( X R Y )  =  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) )
Distinct variable groups:    z, G    z, P    z, X    z, Y
Allowed substitution hints:    ph( z)    B( z)    R( z)    S( z)

Proof of Theorem isray
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isray.3 . . . 4  |-  R  =  (ray `  G )
2 df-ray2 26152 . . . . . 6  |- ray  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } ) ) )
32a1i 10 . . . . 5  |-  ( ph  -> ray  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f ) ,  y  e.  (PPoints `  f
)  |->  if ( x  =/=  y ,  ( ( x ( seg `  f ) y )  u.  { z  e.  (PPoints `  f )  |  y  e.  (
x (btw `  f
) z ) } ) ,  { x } ) ) ) )
4 fveq2 5525 . . . . . . 7  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
54adantl 452 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  (PPoints `  f )  =  (PPoints `  G ) )
6 fveq2 5525 . . . . . . . . . 10  |-  ( f  =  G  ->  ( seg `  f )  =  ( seg `  G
) )
76oveqd 5875 . . . . . . . . 9  |-  ( f  =  G  ->  (
x ( seg `  f
) y )  =  ( x ( seg `  G ) y ) )
8 fveq2 5525 . . . . . . . . . . . 12  |-  ( f  =  G  ->  (btw `  f )  =  (btw
`  G ) )
98oveqd 5875 . . . . . . . . . . 11  |-  ( f  =  G  ->  (
x (btw `  f
) z )  =  ( x (btw `  G ) z ) )
109eleq2d 2350 . . . . . . . . . 10  |-  ( f  =  G  ->  (
y  e.  ( x (btw `  f )
z )  <->  y  e.  ( x (btw `  G ) z ) ) )
114, 10rabeqbidv 2783 . . . . . . . . 9  |-  ( f  =  G  ->  { z  e.  (PPoints `  f
)  |  y  e.  ( x (btw `  f ) z ) }  =  { z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } )
127, 11uneq12d 3330 . . . . . . . 8  |-  ( f  =  G  ->  (
( x ( seg `  f ) y )  u.  { z  e.  (PPoints `  f )  |  y  e.  (
x (btw `  f
) z ) } )  =  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) )
1312ifeq1d 3579 . . . . . . 7  |-  ( f  =  G  ->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } )  =  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  { z  e.  (PPoints `  G )  |  y  e.  (
x (btw `  G
) z ) } ) ,  { x } ) )
1413adantl 452 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } )  =  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  { z  e.  (PPoints `  G )  |  y  e.  (
x (btw `  G
) z ) } ) ,  { x } ) )
155, 5, 14mpt2eq123dv 5910 . . . . 5  |-  ( (
ph  /\  f  =  G )  ->  (
x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  ( ( x ( seg `  f
) y )  u. 
{ z  e.  (PPoints `  f )  |  y  e.  ( x (btw
`  f ) z ) } ) ,  { x } ) )  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } ) ) )
16 isray.5 . . . . 5  |-  ( ph  ->  G  e. Ibg )
17 fvex 5539 . . . . . . 7  |-  (PPoints `  G
)  e.  _V
1817, 17mpt2ex 6198 . . . . . 6  |-  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } ) )  e.  _V
1918a1i 10 . . . . 5  |-  ( ph  ->  ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G
)  |->  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  { z  e.  (PPoints `  G )  |  y  e.  (
x (btw `  G
) z ) } ) ,  { x } ) )  e. 
_V )
203, 15, 16, 19fvmptd 5606 . . . 4  |-  ( ph  ->  (ray `  G )  =  ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  ( ( x ( seg `  G ) y )  u.  {
z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } ) ,  {
x } ) ) )
211, 20syl5eq 2327 . . 3  |-  ( ph  ->  R  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } ) ) )
22 simpl 443 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
23 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
2422, 23neeq12d 2461 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  =/=  y  <->  X  =/=  Y ) )
25 isray.2 . . . . . . . . 9  |-  S  =  ( seg `  G
)
2625eqcomi 2287 . . . . . . . 8  |-  ( seg `  G )  =  S
2726a1i 10 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( seg `  G
)  =  S )
2827, 22, 23oveq123d 5879 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x ( seg `  G ) y )  =  ( X S Y ) )
29 isray.1 . . . . . . . . 9  |-  P  =  (PPoints `  G )
3029eqcomi 2287 . . . . . . . 8  |-  (PPoints `  G
)  =  P
3130a1i 10 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  (PPoints `  G )  =  P )
32 isray.4 . . . . . . . . . . 11  |-  B  =  (btw `  G )
3332eqcomi 2287 . . . . . . . . . 10  |-  (btw `  G )  =  B
3433a1i 10 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  (btw `  G )  =  B )
35 eqidd 2284 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  z  =  z )
3634, 22, 35oveq123d 5879 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x (btw `  G ) z )  =  ( X B z ) )
3723, 36eleq12d 2351 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( y  e.  ( x (btw `  G
) z )  <->  Y  e.  ( X B z ) ) )
3831, 37rabeqbidv 2783 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  { z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) }  =  {
z  e.  P  |  Y  e.  ( X B z ) } )
3928, 38uneq12d 3330 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x ( seg `  G ) y )  u.  {
z  e.  (PPoints `  G
)  |  y  e.  ( x (btw `  G ) z ) } )  =  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) )
40 sneq 3651 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
4140adantr 451 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  { x }  =  { X } )
4224, 39, 41ifbieq12d 3587 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } )  =  if ( X  =/=  Y ,  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } ) )
4342adantl 452 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  if ( x  =/=  y ,  ( ( x ( seg `  G
) y )  u. 
{ z  e.  (PPoints `  G )  |  y  e.  ( x (btw
`  G ) z ) } ) ,  { x } )  =  if ( X  =/=  Y ,  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } ) )
44 isray.6a . . . 4  |-  ( ph  ->  X  e.  P )
4544, 29syl6eleq 2373 . . 3  |-  ( ph  ->  X  e.  (PPoints `  G
) )
46 isray.6b . . . 4  |-  ( ph  ->  Y  e.  P )
4746, 29syl6eleq 2373 . . 3  |-  ( ph  ->  Y  e.  (PPoints `  G
) )
48 ovex 5883 . . . . . 6  |-  ( X S Y )  e. 
_V
4929, 17eqeltri 2353 . . . . . . 7  |-  P  e. 
_V
5049rabex 4165 . . . . . 6  |-  { z  e.  P  |  Y  e.  ( X B z ) }  e.  _V
5148, 50unex 4518 . . . . 5  |-  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } )  e. 
_V
52 snex 4216 . . . . 5  |-  { X }  e.  _V
5351, 52ifex 3623 . . . 4  |-  if ( X  =/=  Y , 
( ( X S Y )  u.  {
z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } )  e.  _V
5453a1i 10 . . 3  |-  ( ph  ->  if ( X  =/= 
Y ,  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } )  e.  _V )
5521, 43, 45, 47, 54ovmpt2d 5975 . 2  |-  ( ph  ->  ( X R Y )  =  if ( X  =/=  Y , 
( ( X S Y )  u.  {
z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } ) )
56 isray.7 . . 3  |-  ( ph  ->  X  =/=  Y )
57 iftrue 3571 . . 3  |-  ( X  =/=  Y  ->  if ( X  =/=  Y ,  ( ( X S Y )  u. 
{ z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } )  =  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) )
5856, 57syl 15 . 2  |-  ( ph  ->  if ( X  =/= 
Y ,  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) ,  { X } )  =  ( ( X S Y )  u. 
{ z  e.  P  |  Y  e.  ( X B z ) } ) )
5955, 58eqtrd 2315 1  |-  ( ph  ->  ( X R Y )  =  ( ( X S Y )  u.  { z  e.  P  |  Y  e.  ( X B z ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    u. cun 3150   ifcif 3565   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  PPointscpoints 26056  btwcbtw 26106  Ibgcibg 26107   segcseg 26130  raycray2 26151
This theorem is referenced by:  segray  26155  rayline  26156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ray2 26152
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