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Theorem outsideofeu 24162
Description: Given a non-degenerate ray, there is a unique point congruent to the segment  B C lying on the ray  A R. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideofeu  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( R  =/=  A  /\  B  =/=  C )  ->  E! x  e.  ( EE `  N ) ( AOutsideOf <. x ,  R >.  /\ 
<. A ,  x >.Cgr <. B ,  C >. ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, N    x, R
Dummy variable  y is distinct from all other variables.

Proof of Theorem outsideofeu
StepHypRef Expression
1 segcon2 24136 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( ( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )
21adantr 453 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  E. x  e.  ( EE `  N
) ( ( R 
Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )
3 simpl1 960 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  N  e.  NN )
4 simpl2l 1010 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
5 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  x  e.  ( EE `  N ) )
6 simpl2r 1011 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  ->  R  e.  ( EE `  N ) )
7 broutsideof2 24153 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  x  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) ) )  -> 
( AOutsideOf <. x ,  R >.  <-> 
( x  =/=  A  /\  R  =/=  A  /\  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) ) )
83, 4, 5, 6, 7syl13anc 1186 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( AOutsideOf <. x ,  R >.  <-> 
( x  =/=  A  /\  R  =/=  A  /\  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) ) )
98adantr 453 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )  ->  ( AOutsideOf
<. x ,  R >.  <->  (
x  =/=  A  /\  R  =/=  A  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) ) )
10 simp3 959 . . . . . . . . . . 11  |-  ( ( x  =/=  A  /\  R  =/=  A  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )
11 simpllr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  B  =/=  C )
1211adantl 454 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  B  =/=  C )
13 simprlr 741 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  <. A ,  x >.Cgr <. B ,  C >. )
14 simp2l 983 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
1514anim1i 553 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( A  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )
16 simpl3 962 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
17 cgrdegen 24035 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  x >.Cgr <. B ,  C >.  ->  ( A  =  x  <->  B  =  C
) ) )
183, 15, 16, 17syl3anc 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  -> 
( <. A ,  x >.Cgr
<. B ,  C >.  -> 
( A  =  x  <-> 
B  =  C ) ) )
1918adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  ( <. A ,  x >.Cgr <. B ,  C >.  -> 
( A  =  x  <-> 
B  =  C ) ) )
2013, 19mpd 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  ( A  =  x  <->  B  =  C ) )
2120necon3bid 2483 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  ( A  =/=  x  <->  B  =/=  C ) )
2212, 21mpbird 225 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  A  =/=  x )
2322necomd 2531 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  x  =/=  A )
24 simplll 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  R  =/=  A )
2524adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  R  =/=  A )
26 simprr 735 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )
2723, 25, 263jca 1134 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( ( R  =/=  A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )  ->  (
x  =/=  A  /\  R  =/=  A  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )
2827expr 600 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )  ->  (
( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. )  ->  ( x  =/= 
A  /\  R  =/=  A  /\  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) ) )
2910, 28impbid2 197 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )  ->  (
( x  =/=  A  /\  R  =/=  A  /\  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  <->  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )
309, 29bitrd 246 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )  ->  ( AOutsideOf
<. x ,  R >.  <->  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) ) )
31 orcom 378 . . . . . . . . 9  |-  ( ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. )  <-> 
( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. ) )
3230, 31syl6bb 254 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. ) )  ->  ( AOutsideOf
<. x ,  R >.  <->  ( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. ) ) )
3332expr 600 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  ( <. A ,  x >.Cgr <. B ,  C >.  -> 
( AOutsideOf <. x ,  R >.  <-> 
( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. ) ) ) )
3433pm5.32rd 623 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  x  e.  ( EE `  N ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( ( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. )  /\  <. A ,  x >.Cgr <. B ,  C >. ) ) )
3534an32s 781 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  /\  x  e.  ( EE `  N
) )  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( ( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. )  /\  <. A ,  x >.Cgr <. B ,  C >. ) ) )
3635rexbidva 2562 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  ( E. x  e.  ( EE `  N ) ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <->  E. x  e.  ( EE `  N ) ( ( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. )  /\  <. A ,  x >.Cgr
<. B ,  C >. ) ) )
372, 36mpbird 225 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  E. x  e.  ( EE `  N
) ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr <. B ,  C >. ) )
38 simpl1 960 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  N  e.  NN )
39 simpl2l 1010 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
40 simpl2r 1011 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  R  e.  ( EE `  N ) )
41 simpl3l 1012 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N ) )
4239, 40, 413jca 1134 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  ( A  e.  ( EE `  N
)  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
43 simpl3r 1013 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
44 simprl 734 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  x  e.  ( EE `  N ) )
45 simprr 735 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  y  e.  ( EE `  N ) )
4643, 44, 453jca 1134 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  ( C  e.  ( EE `  N
)  /\  x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )
4738, 42, 463jca 1134 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  ->  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  x  e.  ( EE `  N
)  /\  y  e.  ( EE `  N ) ) ) )
48 simpr 449 . . . . . . 7  |-  ( ( ( R  =/=  A  /\  B  =/=  C
)  /\  ( ( AOutsideOf
<. x ,  R >.  /\ 
<. A ,  x >.Cgr <. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) ) )  -> 
( ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\ 
<. A ,  y >.Cgr <. B ,  C >. ) ) )
49 outsideofeq 24161 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  x  e.  ( EE `  N
)  /\  y  e.  ( EE `  N ) ) )  ->  (
( ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\ 
<. A ,  y >.Cgr <. B ,  C >. ) )  ->  x  =  y ) )
5049imp 420 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  x  e.  ( EE `  N
)  /\  y  e.  ( EE `  N ) ) )  /\  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) ) )  ->  x  =  y )
5147, 48, 50syl2an 465 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) ) )  /\  ( ( R  =/=  A  /\  B  =/=  C )  /\  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) ) ) )  ->  x  =  y )
5251an4s 801 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  /\  (
( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) )  /\  ( ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) ) ) )  ->  x  =  y )
5352exp32 590 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  (
( x  e.  ( EE `  N )  /\  y  e.  ( EE `  N ) )  ->  ( (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) )  ->  x  =  y ) ) )
5453ralrimivv 2636 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  A. x  e.  ( EE `  N
) A. y  e.  ( EE `  N
) ( ( ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) )  ->  x  =  y ) )
55 opeq1 3798 . . . . . 6  |-  ( x  =  y  ->  <. x ,  R >.  =  <. y ,  R >. )
5655breq2d 4037 . . . . 5  |-  ( x  =  y  ->  ( AOutsideOf
<. x ,  R >.  <->  AOutsideOf <.
y ,  R >. ) )
57 opeq2 3799 . . . . . 6  |-  ( x  =  y  ->  <. A ,  x >.  =  <. A , 
y >. )
5857breq1d 4035 . . . . 5  |-  ( x  =  y  ->  ( <. A ,  x >.Cgr <. B ,  C >.  <->  <. A ,  y >.Cgr <. B ,  C >. ) )
5956, 58anbi12d 693 . . . 4  |-  ( x  =  y  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( AOutsideOf <. y ,  R >.  /\  <. A ,  y
>.Cgr <. B ,  C >. ) ) )
6059reu4 2961 . . 3  |-  ( E! x  e.  ( EE
`  N ) ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( E. x  e.  ( EE `  N
) ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  A. x  e.  ( EE `  N
) A. y  e.  ( EE `  N
) ( ( ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) )  ->  x  =  y ) ) )
6137, 54, 60sylanbrc 647 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( R  =/=  A  /\  B  =/=  C
) )  ->  E! x  e.  ( EE `  N ) ( AOutsideOf <. x ,  R >.  /\ 
<. A ,  x >.Cgr <. B ,  C >. ) )
6261ex 425 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  R  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( R  =/=  A  /\  B  =/=  C )  ->  E! x  e.  ( EE `  N ) ( AOutsideOf <. x ,  R >.  /\ 
<. A ,  x >.Cgr <. B ,  C >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   A.wral 2545   E.wrex 2546   E!wreu 2547   <.cop 3645   class class class wbr 4025   ` cfv 5222   NNcn 9742   EEcee 23924    Btwn cbtwn 23925  Cgrccgr 23926  OutsideOfcoutsideof 24150
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-seq 11042  df-exp 11100  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-sum 12154  df-ee 23927  df-btwn 23928  df-cgr 23929  df-ofs 24014  df-ifs 24070  df-cgr3 24071  df-colinear 24072  df-fs 24073  df-outsideof 24151
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