Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isside1 Unicode version

Theorem isside1 26268
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 26267 . . 3  |-  ( ph  ->  .~  =  { <. x ,  y >.  |  ( x  e.  ( P 
\  M )  /\  y  e.  ( P  \  M )  /\  (
( x S y )  i^i  M )  =  (/) ) } )
87breqd 4050 . 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 2362 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x  e.  ( P  \  M )  <-> 
X  e.  ( P 
\  M ) ) )
13 simpr 447 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  y  =  Y )
1413eleq1d 2362 . . . . 5  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( y  e.  ( P  \  M )  <-> 
Y  e.  ( P 
\  M ) ) )
15 oveq12 5883 . . . . . . 7  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( x S y )  =  ( X S Y ) )
1615ineq1d 3382 . . . . . 6  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ( x S y )  i^i  M
)  =  ( ( X S Y )  i^i  M ) )
1716eqeq1d 2304 . . . . 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 2296 . . . 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 4295 . . 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 1632    e. wcel 1696    \ cdif 3162    i^i cin 3164   (/)c0 3468   class class class wbr 4039   {copab 4092   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Ibgcibg 26210   segcseg 26233  sscsas 26265
This theorem is referenced by:  isside  26269  pdiveql  26271
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
  Copyright terms: Public domain W3C validator