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Theorem outsideofeu 23928
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
StepHypRef Expression
1 segcon2 23902 . . . . 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 963 . . . . . . . . . . . 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 1013 . . . . . . . . . . . 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 1014 . . . . . . . . . . . 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 23919 . . . . . . . . . . . 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 1189 . . . . . . . . . . 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 962 . . . . . . . . . . 11  |-  ( ( x  =/=  A  /\  R  =/=  A  /\  (
x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )
11 simpllr 738 . . . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . . . . . . . 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 965 . . . . . . . . . . . . . . . . . . 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 23801 . . . . . . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . . . . . . 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 2447 . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . 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 1137 . . . . . . . . . . . 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 601 . . . . . . . . . . 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 601 . . . . . . 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 624 . . . . . 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 782 . . . . 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 2524 . . . 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 963 . . . . . . . 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 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 ) ) )  ->  A  e.  ( EE `  N ) )
40 simpl2r 1014 . . . . . . . . 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 1015 . . . . . . . . 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 1137 . . . . . . . 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 1016 . . . . . . . . 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 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 ) ) )  ->  x  e.  ( EE `  N ) )
45 simprr 736 . . . . . . . . 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 1137 . . . . . . . 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 1137 . . . . . . 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 23927 . . . . . . . 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 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 )  /\  y  e.  ( EE `  N ) )  /\  ( ( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  /\  ( AOutsideOf <. y ,  R >.  /\  <. A , 
y >.Cgr <. B ,  C >. ) ) ) )  ->  x  =  y )
5352exp32 591 . . . 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 2596 . . 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 3696 . . . . . 6  |-  ( x  =  y  ->  <. x ,  R >.  =  <. y ,  R >. )
5655breq2d 3932 . . . . 5  |-  ( x  =  y  ->  ( AOutsideOf
<. x ,  R >.  <->  AOutsideOf <.
y ,  R >. ) )
57 opeq2 3697 . . . . . 6  |-  ( x  =  y  ->  <. A ,  x >.  =  <. A , 
y >. )
5857breq1d 3930 . . . . 5  |-  ( x  =  y  ->  ( <. A ,  x >.Cgr <. B ,  C >.  <->  <. A ,  y >.Cgr <. B ,  C >. ) )
5956, 58anbi12d 694 . . . 4  |-  ( x  =  y  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( AOutsideOf <. y ,  R >.  /\  <. A ,  y
>.Cgr <. B ,  C >. ) ) )
6059reu4 2898 . . 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 648 . 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 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   E.wrex 2510   E!wreu 2511   <.cop 3547   class class class wbr 3920   ` cfv 4592   NNcn 9626   EEcee 23690    Btwn cbtwn 23691  Cgrccgr 23692  OutsideOfcoutsideof 23916
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-ee 23693  df-btwn 23694  df-cgr 23695  df-ofs 23780  df-ifs 23836  df-cgr3 23837  df-colinear 23838  df-fs 23839  df-outsideof 23917
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