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Theorem isside0 26267
Description: The predicate "Being on the same side of  L " (For my private use only. Don't use.) (Contributed by FL, 19-Jun-2016.)
Hypotheses
Ref Expression
isside.1  |-  P  =  (PPoints `  G )
isside.2  |-  L  =  (PLines `  G )
isside.3  |-  .~  =  ( (ss `  G ) `
 M )
isside.4  |-  ( ph  ->  G  e. Ibg )
isside.5  |-  ( ph  ->  M  e.  L )
isside.7  |-  S  =  ( seg `  G
)
Assertion
Ref Expression
isside0  |-  ( ph  ->  .~  =  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) } )
Distinct variable groups:    x, y, G    x, M, y    ph, x, y
Allowed substitution hints:    P( x, y)    .~ ( x, y)    S( x, y)    L( x, y)

Proof of Theorem isside0
Dummy variables  g 
l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isside.3 . 2  |-  .~  =  ( (ss `  G ) `
 M )
2 isside.4 . . . . 5  |-  ( ph  ->  G  e. Ibg )
3 fvex 5555 . . . . . 6  |-  (PLines `  G )  e.  _V
43mptex 5762 . . . . 5  |-  ( l  e.  (PLines `  G
)  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } )  e.  _V
5 fveq2 5541 . . . . . . 7  |-  ( g  =  G  ->  (PLines `  g )  =  (PLines `  G ) )
6 fveq2 5541 . . . . . . . . . . 11  |-  ( g  =  G  ->  (PPoints `  g )  =  (PPoints `  G ) )
76difeq1d 3306 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(PPoints `  g )  \ 
l )  =  ( (PPoints `  G )  \  l ) )
87eleq2d 2363 . . . . . . . . 9  |-  ( g  =  G  ->  (
x  e.  ( (PPoints `  g )  \  l
)  <->  x  e.  (
(PPoints `  G )  \ 
l ) ) )
97eleq2d 2363 . . . . . . . . 9  |-  ( g  =  G  ->  (
y  e.  ( (PPoints `  g )  \  l
)  <->  y  e.  ( (PPoints `  G )  \  l ) ) )
10 fveq2 5541 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( seg `  g )  =  ( seg `  G
) )
1110oveqd 5891 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
x ( seg `  g
) y )  =  ( x ( seg `  G ) y ) )
1211ineq1d 3382 . . . . . . . . . 10  |-  ( g  =  G  ->  (
( x ( seg `  g ) y )  i^i  l )  =  ( ( x ( seg `  G ) y )  i^i  l
) )
1312eqeq1d 2304 . . . . . . . . 9  |-  ( g  =  G  ->  (
( ( x ( seg `  g ) y )  i^i  l
)  =  (/)  <->  ( (
x ( seg `  G
) y )  i^i  l )  =  (/) ) )
148, 9, 133anbi123d 1252 . . . . . . . 8  |-  ( g  =  G  ->  (
( x  e.  ( (PPoints `  g )  \  l )  /\  y  e.  ( (PPoints `  g )  \  l
)  /\  ( (
x ( seg `  g
) y )  i^i  l )  =  (/) ) 
<->  ( x  e.  ( (PPoints `  G )  \  l )  /\  y  e.  ( (PPoints `  G )  \  l
)  /\  ( (
x ( seg `  G
) y )  i^i  l )  =  (/) ) ) )
1514opabbidv 4098 . . . . . . 7  |-  ( g  =  G  ->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  g )  \  l
)  /\  y  e.  ( (PPoints `  g )  \  l )  /\  ( ( x ( seg `  g ) y )  i^i  l
)  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } )
165, 15mpteq12dv 4114 . . . . . 6  |-  ( g  =  G  ->  (
l  e.  (PLines `  g )  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  g )  \  l
)  /\  y  e.  ( (PPoints `  g )  \  l )  /\  ( ( x ( seg `  g ) y )  i^i  l
)  =  (/) ) } )  =  ( l  e.  (PLines `  G
)  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } ) )
17 df-sside 26266 . . . . . 6  |- ss  =  ( g  e. Ibg  |->  ( l  e.  (PLines `  g
)  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  g )  \  l
)  /\  y  e.  ( (PPoints `  g )  \  l )  /\  ( ( x ( seg `  g ) y )  i^i  l
)  =  (/) ) } ) )
1816, 17fvmptg 5616 . . . . 5  |-  ( ( G  e. Ibg  /\  (
l  e.  (PLines `  G )  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } )  e.  _V )  ->  (ss `  G )  =  ( l  e.  (PLines `  G )  |->  { <. x ,  y
>.  |  ( x  e.  ( (PPoints `  G
)  \  l )  /\  y  e.  (
(PPoints `  G )  \ 
l )  /\  (
( x ( seg `  G ) y )  i^i  l )  =  (/) ) } ) )
192, 4, 18sylancl 643 . . . 4  |-  ( ph  ->  (ss `  G )  =  ( l  e.  (PLines `  G )  |->  { <. x ,  y
>.  |  ( x  e.  ( (PPoints `  G
)  \  l )  /\  y  e.  (
(PPoints `  G )  \ 
l )  /\  (
( x ( seg `  G ) y )  i^i  l )  =  (/) ) } ) )
2019fveq1d 5543 . . 3  |-  ( ph  ->  ( (ss `  G
) `  M )  =  ( ( l  e.  (PLines `  G
)  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } ) `  M ) )
21 isside.5 . . . . 5  |-  ( ph  ->  M  e.  L )
22 isside.2 . . . . 5  |-  L  =  (PLines `  G )
2321, 22syl6eleq 2386 . . . 4  |-  ( ph  ->  M  e.  (PLines `  G ) )
24 difeq2 3301 . . . . . . . 8  |-  ( l  =  M  ->  (
(PPoints `  G )  \ 
l )  =  ( (PPoints `  G )  \  M ) )
2524eleq2d 2363 . . . . . . 7  |-  ( l  =  M  ->  (
x  e.  ( (PPoints `  G )  \  l
)  <->  x  e.  (
(PPoints `  G )  \  M ) ) )
2624eleq2d 2363 . . . . . . 7  |-  ( l  =  M  ->  (
y  e.  ( (PPoints `  G )  \  l
)  <->  y  e.  ( (PPoints `  G )  \  M ) ) )
27 ineq2 3377 . . . . . . . 8  |-  ( l  =  M  ->  (
( x ( seg `  G ) y )  i^i  l )  =  ( ( x ( seg `  G ) y )  i^i  M
) )
2827eqeq1d 2304 . . . . . . 7  |-  ( l  =  M  ->  (
( ( x ( seg `  G ) y )  i^i  l
)  =  (/)  <->  ( (
x ( seg `  G
) y )  i^i 
M )  =  (/) ) )
2925, 26, 283anbi123d 1252 . . . . . 6  |-  ( l  =  M  ->  (
( x  e.  ( (PPoints `  G )  \  l )  /\  y  e.  ( (PPoints `  G )  \  l
)  /\  ( (
x ( seg `  G
) y )  i^i  l )  =  (/) ) 
<->  ( x  e.  ( (PPoints `  G )  \  M )  /\  y  e.  ( (PPoints `  G
)  \  M )  /\  ( ( x ( seg `  G ) y )  i^i  M
)  =  (/) ) ) )
3029opabbidv 4098 . . . . 5  |-  ( l  =  M  ->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  M
)  /\  y  e.  ( (PPoints `  G )  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } )
31 eqid 2296 . . . . 5  |-  ( l  e.  (PLines `  G
)  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } )  =  ( l  e.  (PLines `  G
)  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } )
32 3anass 938 . . . . . . . 8  |-  ( ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) )  <->  ( x  e.  ( (PPoints `  G
)  \  l )  /\  ( y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) ) )
3332opabbii 4099 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  ( y  e.  ( (PPoints `  G
)  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) ) }
3433a1i 10 . . . . . 6  |-  ( l  e.  (PLines `  G
)  ->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  ( y  e.  ( (PPoints `  G
)  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) ) } )
35 fvex 5555 . . . . . . . 8  |-  (PPoints `  G
)  e.  _V
36 difexg 4178 . . . . . . . 8  |-  ( (PPoints `  G )  e.  _V  ->  ( (PPoints `  G
)  \  l )  e.  _V )
3735, 36ax-mp 8 . . . . . . 7  |-  ( (PPoints `  G )  \  l
)  e.  _V
3837zfausab 4179 . . . . . . . 8  |-  { y  |  ( y  e.  ( (PPoints `  G
)  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) }  e.  _V
3938a1i 10 . . . . . . 7  |-  ( x  e.  ( (PPoints `  G
)  \  l )  ->  { y  |  ( y  e.  ( (PPoints `  G )  \  l
)  /\  ( (
x ( seg `  G
) y )  i^i  l )  =  (/) ) }  e.  _V )
4037, 39opabex3 5785 . . . . . 6  |-  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  ( y  e.  ( (PPoints `  G
)  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) ) }  e.  _V
4134, 40syl6eqel 2384 . . . . 5  |-  ( l  e.  (PLines `  G
)  ->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) }  e.  _V )
4230, 31, 41fvmpt3 5620 . . . 4  |-  ( M  e.  (PLines `  G
)  ->  ( (
l  e.  (PLines `  G )  |->  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  l
)  /\  y  e.  ( (PPoints `  G )  \  l )  /\  ( ( x ( seg `  G ) y )  i^i  l
)  =  (/) ) } ) `  M )  =  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G )  \  M
)  /\  y  e.  ( (PPoints `  G )  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } )
4323, 42syl 15 . . 3  |-  ( ph  ->  ( ( l  e.  (PLines `  G )  |->  { <. x ,  y
>.  |  ( x  e.  ( (PPoints `  G
)  \  l )  /\  y  e.  (
(PPoints `  G )  \ 
l )  /\  (
( x ( seg `  G ) y )  i^i  l )  =  (/) ) } ) `  M )  =  { <. x ,  y >.  |  ( x  e.  ( (PPoints `  G
)  \  M )  /\  y  e.  (
(PPoints `  G )  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) } )
44 isside.1 . . . . . . . . 9  |-  P  =  (PPoints `  G )
4544eqcomi 2300 . . . . . . . 8  |-  (PPoints `  G
)  =  P
4645difeq1i 3303 . . . . . . 7  |-  ( (PPoints `  G )  \  M
)  =  ( P 
\  M )
4746a1i 10 . . . . . 6  |-  ( ph  ->  ( (PPoints `  G
)  \  M )  =  ( P  \  M ) )
4847eleq2d 2363 . . . . 5  |-  ( ph  ->  ( x  e.  ( (PPoints `  G )  \  M )  <->  x  e.  ( P  \  M ) ) )
4947eleq2d 2363 . . . . 5  |-  ( ph  ->  ( y  e.  ( (PPoints `  G )  \  M )  <->  y  e.  ( P  \  M ) ) )
50 isside.7 . . . . . . . . . 10  |-  S  =  ( seg `  G
)
5150eqcomi 2300 . . . . . . . . 9  |-  ( seg `  G )  =  S
5251a1i 10 . . . . . . . 8  |-  ( ph  ->  ( seg `  G
)  =  S )
5352oveqd 5891 . . . . . . 7  |-  ( ph  ->  ( x ( seg `  G ) y )  =  ( x S y ) )
5453ineq1d 3382 . . . . . 6  |-  ( ph  ->  ( ( x ( seg `  G ) y )  i^i  M
)  =  ( ( x S y )  i^i  M ) )
5554eqeq1d 2304 . . . . 5  |-  ( ph  ->  ( ( ( x ( seg `  G
) y )  i^i 
M )  =  (/)  <->  (
( x S y )  i^i  M )  =  (/) ) )
5648, 49, 553anbi123d 1252 . . . 4  |-  ( ph  ->  ( ( x  e.  ( (PPoints `  G
)  \  M )  /\  y  e.  (
(PPoints `  G )  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) )  <->  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i 
M )  =  (/) ) ) )
5756opabbidv 4098 . . 3  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  ( (PPoints `  G
)  \  M )  /\  y  e.  (
(PPoints `  G )  \  M )  /\  (
( x ( seg `  G ) y )  i^i  M )  =  (/) ) }  =  { <. x ,  y >.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i 
M )  =  (/) ) } )
5820, 43, 573eqtrd 2332 . 2  |-  ( ph  ->  ( (ss `  G
) `  M )  =  { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i 
M )  =  (/) ) } )
591, 58syl5eq 2340 1  |-  ( ph  ->  .~  =  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    \ cdif 3162    i^i cin 3164   (/)c0 3468   {copab 4092    e. cmpt 4093   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Ibgcibg 26210   segcseg 26233  sscsas 26265
This theorem is referenced by:  isside1  26268  bosser  26270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-sside 26266
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