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Theorem outsideofeu 28162
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

Proof of Theorem outsideofeu
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 segcon2 28136 . . . . 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 465 . . . 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 991 . . . . . . . . . . . 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 1041 . . . . . . . . . . . 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 461 . . . . . . . . . . . 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 1042 . . . . . . . . . . . 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 28153 . . . . . . . . . . . 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 1220 . . . . . . . . . . 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 465 . . . . . . . . . 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 990 . . . . . . . . . . 11  |-  ( ( x  =/=  A  /\  R  =/=  A  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )
11 simpllr 758 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  B  =/=  C )
1211adantl 466 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . . . . . . . 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 568 . . . . . . . . . . . . . . . . . . 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 993 . . . . . . . . . . . . . . . . . . 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 28035 . . . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . 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 2643 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 2695 . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  R  =/=  A )
2524adantl 466 . . . . . . . . . . . . 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 756 . . . . . . . . . . . . 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 1168 . . . . . . . . . . . 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 615 . . . . . . . . . . 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 204 . . . . . . . . . 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 253 . . . . . . . . 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 387 . . . . . . . . 9  |-  ( ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. )  <-> 
( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. ) )
3230, 31syl6bb 261 . . . . . . . 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 615 . . . . . . 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 640 . . . . . 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 802 . . . . 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 2732 . . . 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 232 . . 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 991 . . . . . . . 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 1041 . . . . . . . . 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 1042 . . . . . . . . 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 1043 . . . . . . . . 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 1168 . . . . . . . 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 1044 . . . . . . . . 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 755 . . . . . . . . 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 756 . . . . . . . . 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 1168 . . . . . . . 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 1168 . . . . . . 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 461 . . . . . . 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 28161 . . . . . . . 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 429 . . . . . . 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 477 . . . . . 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 822 . . . . 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 605 . . . 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 2807 . . 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 4059 . . . . . 6  |-  ( x  =  y  ->  <. x ,  R >.  =  <. y ,  R >. )
5655breq2d 4304 . . . . 5  |-  ( x  =  y  ->  ( AOutsideOf
<. x ,  R >.  <->  AOutsideOf <.
y ,  R >. ) )
57 opeq2 4060 . . . . . 6  |-  ( x  =  y  ->  <. A ,  x >.  =  <. A , 
y >. )
5857breq1d 4302 . . . . 5  |-  ( x  =  y  ->  ( <. A ,  x >.Cgr <. B ,  C >.  <->  <. A ,  y >.Cgr <. B ,  C >. ) )
5956, 58anbi12d 710 . . . 4  |-  ( x  =  y  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( AOutsideOf <. y ,  R >.  /\  <. A ,  y
>.Cgr <. B ,  C >. ) ) )
6059reu4 3153 . . 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 664 . 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 434 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   E!wreu 2717   <.cop 3883   class class class wbr 4292   ` cfv 5418   NNcn 10322   EEcee 23134    Btwn cbtwn 23135  Cgrccgr 23136  OutsideOfcoutsideof 28150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-ee 23137  df-btwn 23138  df-cgr 23139  df-ofs 28014  df-colinear 28070  df-ifs 28071  df-cgr3 28072  df-fs 28073  df-outsideof 28151
This theorem is referenced by: (None)
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