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Theorem isside1 26165
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
)
isside1.8  |-  ( ph  ->  X  e.  A )
isside1.9  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
isside1  |-  ( ph  ->  ( X  .~  Y  <->  ( X  e.  ( P 
\  M )  /\  Y  e.  ( P  \  M )  /\  (
( X S Y )  i^i  M )  =  (/) ) ) )

Proof of Theorem isside1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isside.1 . . . 4  |-  P  =  (PPoints `  G )
2 isside.2 . . . 4  |-  L  =  (PLines `  G )
3 isside.3 . . . 4  |-  .~  =  ( (ss `  G ) `
 M )
4 isside.4 . . . 4  |-  ( ph  ->  G  e. Ibg )
5 isside.5 . . . 4  |-  ( ph  ->  M  e.  L )
6 isside.7 . . . 4  |-  S  =  ( seg `  G
)
71, 2, 3, 4, 5, 6isside0 26164 . . 3  |-  ( ph  ->  .~  =  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) } )
87breqd 4034 . 2  |-  ( ph  ->  ( X  .~  Y  <->  X { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i 
M )  =  (/) ) } Y ) )
9 isside1.8 . . 3  |-  ( ph  ->  X  e.  A )
10 isside1.9 . . 3  |-  ( ph  ->  Y  e.  B )
11 simpl 443 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  x  =  X )
1211eleq1d 2349 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  ( P  \  M )  <-> 
X  e.  ( P 
\  M ) ) )
13 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1413eleq1d 2349 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( y  e.  ( P  \  M )  <-> 
Y  e.  ( P 
\  M ) ) )
15 oveq12 5867 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x S y )  =  ( X S Y ) )
1615ineq1d 3369 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x S y )  i^i  M
)  =  ( ( X S Y )  i^i  M ) )
1716eqeq1d 2291 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( ( x S y )  i^i 
M )  =  (/)  <->  (
( X S Y )  i^i  M )  =  (/) ) )
1812, 14, 173anbi123d 1252 . . . 4  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i 
M )  =  (/) ) 
<->  ( X  e.  ( P  \  M )  /\  Y  e.  ( P  \  M )  /\  ( ( X S Y )  i^i 
M )  =  (/) ) ) )
19 eqid 2283 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) }  =  { <. x ,  y
>.  |  ( x  e.  ( P  \  M
)  /\  y  e.  ( P  \  M )  /\  ( ( x S y )  i^i 
M )  =  (/) ) }
2018, 19brabga 4279 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) } Y  <->  ( X  e.  ( P 
\  M )  /\  Y  e.  ( P  \  M )  /\  (
( X S Y )  i^i  M )  =  (/) ) ) )
219, 10, 20syl2anc 642 . 2  |-  ( ph  ->  ( X { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) } Y  <->  ( X  e.  ( P 
\  M )  /\  Y  e.  ( P  \  M )  /\  (
( X S Y )  i^i  M )  =  (/) ) ) )
228, 21bitrd 244 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151   (/)c0 3455   class class class wbr 4023   {copab 4076   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Ibgcibg 26107   segcseg 26130  sscsas 26162
This theorem is referenced by:  isside  26166  pdiveql  26168
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-sside 26163
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