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Theorem outsideofeu 28091
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 28065 . . . . 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 462 . . . 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 986 . . . . . . . . . . . 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 1036 . . . . . . . . . . . 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 458 . . . . . . . . . . . 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 1037 . . . . . . . . . . . 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 28082 . . . . . . . . . . . 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 1215 . . . . . . . . . . 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 462 . . . . . . . . . 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 985 . . . . . . . . . . 11  |-  ( ( x  =/=  A  /\  R  =/=  A  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )
11 simpllr 753 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  B  =/=  C )
1211adantl 463 . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . . . . . . 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 565 . . . . . . . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . . . . . . 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 27964 . . . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 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 2641 . . . . . . . . . . . . . . 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 2693 . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  R  =/=  A )
2524adantl 463 . . . . . . . . . . . . 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 751 . . . . . . . . . . . . 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 1163 . . . . . . . . . . . 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 612 . . . . . . . . . . 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 612 . . . . . . 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 635 . . . . . 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 797 . . . . 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 2730 . . . 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 986 . . . . . . . 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 1036 . . . . . . . . 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 1037 . . . . . . . . 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 1038 . . . . . . . . 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 1163 . . . . . . . 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 1039 . . . . . . . . 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 750 . . . . . . . . 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 751 . . . . . . . . 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 1163 . . . . . . . 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 1163 . . . . . . 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 458 . . . . . . 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 28090 . . . . . . . 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 474 . . . . . 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 817 . . . . 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 602 . . . 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 2805 . . 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 4056 . . . . . 6  |-  ( x  =  y  ->  <. x ,  R >.  =  <. y ,  R >. )
5655breq2d 4301 . . . . 5  |-  ( x  =  y  ->  ( AOutsideOf
<. x ,  R >.  <->  AOutsideOf <.
y ,  R >. ) )
57 opeq2 4057 . . . . . 6  |-  ( x  =  y  ->  <. A ,  x >.  =  <. A , 
y >. )
5857breq1d 4299 . . . . 5  |-  ( x  =  y  ->  ( <. A ,  x >.Cgr <. B ,  C >.  <->  <. A ,  y >.Cgr <. B ,  C >. ) )
5956, 58anbi12d 705 . . . 4  |-  ( x  =  y  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( AOutsideOf <. y ,  R >.  /\  <. A ,  y
>.Cgr <. B ,  C >. ) ) )
6059reu4 3150 . . 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 659 . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   E!wreu 2715   <.cop 3880   class class class wbr 4289   ` cfv 5415   NNcn 10318   EEcee 23069    Btwn cbtwn 23070  Cgrccgr 23071  OutsideOfcoutsideof 28079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-ee 23072  df-btwn 23073  df-cgr 23074  df-ofs 27943  df-colinear 27999  df-ifs 28000  df-cgr3 28001  df-fs 28002  df-outsideof 28080
This theorem is referenced by: (None)
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