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Theorem oppperpex 24782
Description: Restating colperpex 24762 using the "opposite side of a line" relation. (Contributed by Thierry Arnoux, 2-Aug-2020.)
Hypotheses
Ref Expression
hpg.p  |-  P  =  ( Base `  G
)
hpg.d  |-  .-  =  ( dist `  G )
hpg.i  |-  I  =  (Itv `  G )
hpg.o  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
opphl.l  |-  L  =  (LineG `  G )
opphl.d  |-  ( ph  ->  D  e.  ran  L
)
opphl.g  |-  ( ph  ->  G  e. TarskiG )
opphl.k  |-  K  =  (hlG `  G )
oppperpex.1  |-  ( ph  ->  A  e.  D )
oppperpex.2  |-  ( ph  ->  C  e.  P )
oppperpex.3  |-  ( ph  ->  -.  C  e.  D
)
oppperpex.4  |-  ( ph  ->  GDimTarskiG 2 )
Assertion
Ref Expression
oppperpex  |-  ( ph  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) D  /\  C O p ) )
Distinct variable groups:    D, a,
b    I, a, b    P, a, b    A, p, t    D, p, t    C, p, t    G, p, t    t, L    I, p, t    K, p, t    t, O    P, p, t    ph, p, t    .- , p, t    t, a, b    L, p
Allowed substitution hints:    ph( a, b)    A( a, b)    C( a, b)    G( a, b)    K( a, b)    L( a, b)    .- ( a, b)    O( p, a, b)

Proof of Theorem oppperpex
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simprrl 772 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  ( A L p ) (⟂G `  G ) ( A L x ) )
2 hpg.p . . . . . . 7  |-  P  =  ( Base `  G
)
3 hpg.i . . . . . . 7  |-  I  =  (Itv `  G )
4 opphl.l . . . . . . 7  |-  L  =  (LineG `  G )
5 opphl.g . . . . . . . 8  |-  ( ph  ->  G  e. TarskiG )
65ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  G  e. TarskiG )
7 opphl.d . . . . . . . . 9  |-  ( ph  ->  D  e.  ran  L
)
87ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  D  e.  ran  L )
9 oppperpex.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  D )
109ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  A  e.  D )
112, 4, 3, 6, 8, 10tglnpt 24581 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  A  e.  P )
12 simplr 760 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  x  e.  D )
132, 4, 3, 6, 8, 12tglnpt 24581 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  x  e.  P )
14 simpr 462 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  A  =/=  x )
152, 3, 4, 6, 11, 13, 14, 14, 8, 10, 12tglinethru 24668 . . . . . 6  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  D  =  ( A L x ) )
1615adantr 466 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  D  =  ( A L x ) )
171, 16breqtrrd 4447 . . . 4  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  ( A L p ) (⟂G `  G ) D )
18 oppperpex.3 . . . . . . 7  |-  ( ph  ->  -.  C  e.  D
)
1918ad3antrrr 734 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  -.  C  e.  D )
20 hpg.d . . . . . . 7  |-  .-  =  ( dist `  G )
216adantr 466 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  G  e. TarskiG )
228adantr 466 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  D  e.  ran  L )
2310adantr 466 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  A  e.  D )
24 simprl 762 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  p  e.  P )
252, 20, 3, 4, 21, 22, 23, 24, 17footne 24752 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  -.  p  e.  D )
2614ad3antrrr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  ->  A  =/=  x )
2726neneqd 2625 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  ->  -.  A  =  x
)
28 simprrl 772 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
( t  e.  ( A L x )  \/  A  =  x ) )
2928orcomd 389 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
( A  =  x  \/  t  e.  ( A L x ) ) )
3029ord 378 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
( -.  A  =  x  ->  t  e.  ( A L x ) ) )
3127, 30mpd 15 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
t  e.  ( A L x ) )
3215ad3antrrr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  ->  D  =  ( A L x ) )
3331, 32eleqtrrd 2513 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
t  e.  D )
34 simprrr 773 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
t  e.  ( C I p ) )
3533, 34jca 534 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  ( t  e.  P  /\  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  -> 
( t  e.  D  /\  t  e.  ( C I p ) ) )
3635ex 435 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  ->  ( ( t  e.  P  /\  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) )  -> 
( t  e.  D  /\  t  e.  ( C I p ) ) ) )
3736reximdv2 2896 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  ->  ( E. t  e.  P  ( (
t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) )  ->  E. t  e.  D  t  e.  ( C I p ) ) )
3837imp 430 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  E. t  e.  P  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) )  ->  E. t  e.  D  t  e.  ( C I p ) )
3938anasss 651 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  ->  E. t  e.  D  t  e.  ( C I p ) )
4039anasss 651 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  E. t  e.  D  t  e.  ( C I p ) )
4119, 25, 40jca31 536 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  ( ( -.  C  e.  D  /\  -.  p  e.  D
)  /\  E. t  e.  D  t  e.  ( C I p ) ) )
42 hpg.o . . . . . . . . 9  |-  O  =  { <. a ,  b
>.  |  ( (
a  e.  ( P 
\  D )  /\  b  e.  ( P  \  D ) )  /\  E. t  e.  D  t  e.  ( a I b ) ) }
43 oppperpex.2 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  P )
4443ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  C  e.  P )
4544ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  ->  C  e.  P
)
46 simplr 760 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  ->  p  e.  P
)
472, 20, 3, 42, 45, 46islnopp 24768 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  ->  ( C O p  <->  ( ( -.  C  e.  D  /\  -.  p  e.  D
)  /\  E. t  e.  D  t  e.  ( C I p ) ) ) )
4847adantr 466 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  x  e.  D )  /\  A  =/=  x )  /\  p  e.  P )  /\  ( A L p ) (⟂G `  G ) ( A L x ) )  /\  E. t  e.  P  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) )  -> 
( C O p  <-> 
( ( -.  C  e.  D  /\  -.  p  e.  D )  /\  E. t  e.  D  t  e.  ( C I p ) ) ) )
4948anasss 651 . . . . . 6  |-  ( ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  p  e.  P )  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )  ->  ( C O p  <->  ( ( -.  C  e.  D  /\  -.  p  e.  D
)  /\  E. t  e.  D  t  e.  ( C I p ) ) ) )
5049anasss 651 . . . . 5  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  ( C O p  <->  ( ( -.  C  e.  D  /\  -.  p  e.  D
)  /\  E. t  e.  D  t  e.  ( C I p ) ) ) )
5141, 50mpbird 235 . . . 4  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  C O p )
5217, 51jca 534 . . 3  |-  ( ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x
)  /\  ( p  e.  P  /\  (
( A L p ) (⟂G `  G
) ( A L x )  /\  E. t  e.  P  (
( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) ) )  ->  ( ( A L p ) (⟂G `  G ) D  /\  C O p ) )
53 oppperpex.4 . . . . 5  |-  ( ph  ->  GDimTarskiG 2 )
5453ad2antrr 730 . . . 4  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  GDimTarskiG 2
)
552, 20, 3, 4, 6, 11, 13, 44, 14, 54colperpex 24762 . . 3  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G ) ( A L x )  /\  E. t  e.  P  ( ( t  e.  ( A L x )  \/  A  =  x )  /\  t  e.  ( C I p ) ) ) )
5652, 55reximddv 2901 . 2  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =/=  x )  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G ) D  /\  C O p ) )
572, 3, 4, 5, 7, 9tglnpt2 24673 . 2  |-  ( ph  ->  E. x  e.  D  A  =/=  x )
5856, 57r19.29a 2970 1  |-  ( ph  ->  E. p  e.  P  ( ( A L p ) (⟂G `  G
) D  /\  C O p ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776    \ cdif 3433   class class class wbr 4420   {copab 4478   ran crn 4851   ` cfv 5598  (class class class)co 6302   2c2 10660   Basecbs 15109   distcds 15187  TarskiGcstrkg 24465  DimTarskiGcstrkgld 24469  Itvcitv 24471  LineGclng 24472  hlGchlg 24632  ⟂Gcperpg 24727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-hash 12516  df-word 12657  df-concat 12659  df-s1 12660  df-s2 12935  df-s3 12936  df-trkgc 24483  df-trkgb 24484  df-trkgcb 24485  df-trkgld 24487  df-trkg 24488  df-cgrg 24543  df-leg 24615  df-mir 24685  df-rag 24726  df-perpg 24728
This theorem is referenced by:  lnperpex  24832
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