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Theorem sgplpte21 26132
Description: The predicate "is a non-degenerated segment". (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
sgplpte.1  |-  P  =  (PPoints `  G )
sgplpte.3  |-  S  =  ( seg `  G
)
sgplpte.4  |-  ( ph  ->  G  e. Ibg )
sgplpte.5  |-  ( ph  ->  X  e.  P )
sgplpte21.2  |-  B  =  (btw `  G )
sgplpte21.6  |-  ( ph  ->  Y  e.  P )
sgplpte21.7  |-  ( ph  ->  X  =/=  Y )
Assertion
Ref Expression
sgplpte21  |-  ( ph  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
Distinct variable groups:    z, G    ph, z    z, P    z, X    z, Y
Allowed substitution hints:    B( z)    S( z)

Proof of Theorem sgplpte21
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sgplpte.3 . . . 4  |-  S  =  ( seg `  G
)
21a1i 10 . . 3  |-  ( ph  ->  S  =  ( seg `  G ) )
32oveqd 5875 . 2  |-  ( ph  ->  ( X S Y )  =  ( X ( seg `  G
) Y ) )
4 df-seg2 26131 . . . . . . 7  |-  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
54a1i 10 . . . . . 6  |-  ( ph  ->  seg  =  ( f  e. Ibg  |->  ( x  e.  (PPoints `  f ) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
) ) ) )
6 fveq2 5525 . . . . . . . 8  |-  ( f  =  G  ->  (PPoints `  f )  =  (PPoints `  G ) )
76adantl 452 . . . . . . 7  |-  ( (
ph  /\  f  =  G )  ->  (PPoints `  f )  =  (PPoints `  G ) )
8 fveq2 5525 . . . . . . . . . . . . 13  |-  ( f  =  G  ->  (btw `  f )  =  (btw
`  G ) )
98adantl 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  f  =  G )  ->  (btw `  f )  =  (btw
`  G ) )
109oveqd 5875 . . . . . . . . . . 11  |-  ( (
ph  /\  f  =  G )  ->  (
x (btw `  f
) y )  =  ( x (btw `  G ) y ) )
1110eleq2d 2350 . . . . . . . . . 10  |-  ( (
ph  /\  f  =  G )  ->  (
z  e.  ( x (btw `  f )
y )  <->  z  e.  ( x (btw `  G ) y ) ) )
12 biidd 228 . . . . . . . . . 10  |-  ( (
ph  /\  f  =  G )  ->  (
z  =  x  <->  z  =  x ) )
13 biidd 228 . . . . . . . . . 10  |-  ( (
ph  /\  f  =  G )  ->  (
z  =  y  <->  z  =  y ) )
1411, 12, 133orbi123d 1251 . . . . . . . . 9  |-  ( (
ph  /\  f  =  G )  ->  (
( z  e.  ( x (btw `  f
) y )  \/  z  =  x  \/  z  =  y )  <-> 
( z  e.  ( x (btw `  G
) y )  \/  z  =  x  \/  z  =  y ) ) )
157, 14rabeqbidv 2783 . . . . . . . 8  |-  ( (
ph  /\  f  =  G )  ->  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) }  =  { z  e.  (PPoints `  G )  |  ( z  e.  ( x (btw `  G )
y )  \/  z  =  x  \/  z  =  y ) } )
1615ifeq1d 3579 . . . . . . 7  |-  ( (
ph  /\  f  =  G )  ->  if ( x  =/=  y ,  { z  e.  (PPoints `  f )  |  ( z  e.  ( x (btw `  f )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
)  =  if ( x  =/=  y ,  { z  e.  (PPoints `  G )  |  ( z  e.  ( x (btw `  G )
y )  \/  z  =  x  \/  z  =  y ) } ,  { x }
) )
177, 7, 16mpt2eq123dv 5910 . . . . . 6  |-  ( (
ph  /\  f  =  G )  ->  (
x  e.  (PPoints `  f
) ,  y  e.  (PPoints `  f )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  f
)  |  ( z  e.  ( x (btw
`  f ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
18 sgplpte.4 . . . . . 6  |-  ( ph  ->  G  e. Ibg )
19 fvex 5539 . . . . . . . 8  |-  (PPoints `  G
)  e.  _V
2019, 19mpt2ex 6198 . . . . . . 7  |-  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  e.  _V
2120a1i 10 . . . . . 6  |-  ( ph  ->  ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G
)  |->  if ( x  =/=  y ,  {
z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  e.  _V )
225, 17, 18, 21fvmptd 5606 . . . . 5  |-  ( ph  ->  ( seg `  G
)  =  ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
2322oveqd 5875 . . . 4  |-  ( ph  ->  ( X ( seg `  G ) Y )  =  ( X ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) Y ) )
24 sgplpte.1 . . . . . . . 8  |-  P  =  (PPoints `  G )
2524eqcomi 2287 . . . . . . 7  |-  (PPoints `  G
)  =  P
2625a1i 10 . . . . . 6  |-  ( ph  ->  (PPoints `  G )  =  P )
27 sgplpte21.2 . . . . . . . . . . . . 13  |-  B  =  (btw `  G )
2827eqcomi 2287 . . . . . . . . . . . 12  |-  (btw `  G )  =  B
2928a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  (btw `  G )  =  B )
3029oveqd 5875 . . . . . . . . . 10  |-  ( ph  ->  ( x (btw `  G ) y )  =  ( x B y ) )
3130eleq2d 2350 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  ( x (btw `  G
) y )  <->  z  e.  ( x B y ) ) )
32 biidd 228 . . . . . . . . 9  |-  ( ph  ->  ( z  =  x  <-> 
z  =  x ) )
33 biidd 228 . . . . . . . . 9  |-  ( ph  ->  ( z  =  y  <-> 
z  =  y ) )
3431, 32, 333orbi123d 1251 . . . . . . . 8  |-  ( ph  ->  ( ( z  e.  ( x (btw `  G ) y )  \/  z  =  x  \/  z  =  y )  <->  ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y ) ) )
3526, 34rabeqbidv 2783 . . . . . . 7  |-  ( ph  ->  { z  e.  (PPoints `  G )  |  ( z  e.  ( x (btw `  G )
y )  \/  z  =  x  \/  z  =  y ) }  =  { z  e.  P  |  ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y ) } )
3635ifeq1d 3579 . . . . . 6  |-  ( ph  ->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } )  =  if ( x  =/=  y ,  {
z  e.  P  | 
( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y ) } ,  {
x } ) )
3726, 26, 36mpt2eq123dv 5910 . . . . 5  |-  ( ph  ->  ( x  e.  (PPoints `  G ) ,  y  e.  (PPoints `  G
)  |->  if ( x  =/=  y ,  {
z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) )  =  ( x  e.  P ,  y  e.  P  |->  if ( x  =/=  y ,  { z  e.  P  |  ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) )
38 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
39 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
4038, 39neeq12d 2461 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x  =/=  y  <->  X  =/=  Y ) )
41 oveq12 5867 . . . . . . . . . 10  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x B y )  =  ( X B Y ) )
4241eleq2d 2350 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  e.  ( x B y )  <-> 
z  e.  ( X B Y ) ) )
43 simpl 443 . . . . . . . . . 10  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
4443eqeq2d 2294 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  =  x  <-> 
z  =  X ) )
45 simpr 447 . . . . . . . . . 10  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
4645eqeq2d 2294 . . . . . . . . 9  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( z  =  y  <-> 
z  =  Y ) )
4742, 44, 463orbi123d 1251 . . . . . . . 8  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y )  <->  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) )
4847adantl 452 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y )  <->  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) ) )
4948rabbidv 2780 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  { z  e.  P  |  ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y ) }  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
50 sneq 3651 . . . . . . 7  |-  ( x  =  X  ->  { x }  =  { X } )
5150ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  { x }  =  { X } )
5240, 49, 51ifbieq12d 3587 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  if ( x  =/=  y ,  { z  e.  P  |  ( z  e.  ( x B y )  \/  z  =  x  \/  z  =  y ) } ,  { x } )  =  if ( X  =/=  Y ,  {
z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } ) )
53 sgplpte.5 . . . . 5  |-  ( ph  ->  X  e.  P )
54 sgplpte21.6 . . . . 5  |-  ( ph  ->  Y  e.  P )
5524, 19eqeltri 2353 . . . . . . . 8  |-  P  e. 
_V
5655rabex 4165 . . . . . . 7  |-  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) }  e.  _V
57 snex 4216 . . . . . . 7  |-  { X }  e.  _V
5856, 57ifex 3623 . . . . . 6  |-  if ( X  =/=  Y ,  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } )  e. 
_V
5958a1i 10 . . . . 5  |-  ( ph  ->  if ( X  =/= 
Y ,  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } )  e.  _V )
6037, 52, 53, 54, 59ovmpt2d 5975 . . . 4  |-  ( ph  ->  ( X ( x  e.  (PPoints `  G
) ,  y  e.  (PPoints `  G )  |->  if ( x  =/=  y ,  { z  e.  (PPoints `  G
)  |  ( z  e.  ( x (btw
`  G ) y )  \/  z  =  x  \/  z  =  y ) } ,  { x } ) ) Y )  =  if ( X  =/= 
Y ,  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } ) )
6123, 60eqtrd 2315 . . 3  |-  ( ph  ->  ( X ( seg `  G ) Y )  =  if ( X  =/=  Y ,  {
z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } ) )
62 sgplpte21.7 . . . 4  |-  ( ph  ->  X  =/=  Y )
63 iftrue 3571 . . . 4  |-  ( X  =/=  Y  ->  if ( X  =/=  Y ,  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
6462, 63syl 15 . . 3  |-  ( ph  ->  if ( X  =/= 
Y ,  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } ,  { X } )  =  {
z  e.  P  | 
( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
6561, 64eqtrd 2315 . 2  |-  ( ph  ->  ( X ( seg `  G ) Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
663, 65eqtrd 2315 1  |-  ( ph  ->  ( X S Y )  =  { z  e.  P  |  ( z  e.  ( X B Y )  \/  z  =  X  \/  z  =  Y ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   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
This theorem is referenced by:  sgplpte21a  26133  xsyysx  26145  bsstrs  26146  nbssntrs  26147
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-3or 935  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-seg2 26131
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