Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  outsideofeu Unicode version

Theorem outsideofeu 26008
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 25982 . . . . 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 452 . . . 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 448 . . . . . . . . . . . 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 25999 . . . . . . . . . . . 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 452 . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  B  =/=  C )
1211adantl 453 . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . . 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 552 . . . . . . . . . . . . . . . . . . 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 25881 . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . 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 2628 . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  =/= 
A  /\  B  =/=  C )  /\  <. A ,  x >.Cgr <. B ,  C >. )  /\  ( x 
Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. ) )  ->  R  =/=  A )
2524adantl 453 . . . . . . . . . . . . 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 734 . . . . . . . . . . . . 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 599 . . . . . . . . . . 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 196 . . . . . . . . . 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 245 . . . . . . . . 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 377 . . . . . . . . 9  |-  ( ( x  Btwn  <. A ,  R >.  \/  R  Btwn  <. A ,  x >. )  <-> 
( R  Btwn  <. A ,  x >.  \/  x  Btwn  <. A ,  R >. ) )
3230, 31syl6bb 253 . . . . . . . 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 599 . . . . . . 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 622 . . . . . 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 780 . . . . 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 2709 . . . 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 224 . . 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 733 . . . . . . . . 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 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 ) ) )  ->  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 448 . . . . . . 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 26007 . . . . . . . 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 419 . . . . . . 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 464 . . . . . 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 800 . . . . 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 589 . . . 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 2784 . . 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 3971 . . . . . 6  |-  ( x  =  y  ->  <. x ,  R >.  =  <. y ,  R >. )
5655breq2d 4211 . . . . 5  |-  ( x  =  y  ->  ( AOutsideOf
<. x ,  R >.  <->  AOutsideOf <.
y ,  R >. ) )
57 opeq2 3972 . . . . . 6  |-  ( x  =  y  ->  <. A ,  x >.  =  <. A , 
y >. )
5857breq1d 4209 . . . . 5  |-  ( x  =  y  ->  ( <. A ,  x >.Cgr <. B ,  C >.  <->  <. A ,  y >.Cgr <. B ,  C >. ) )
5956, 58anbi12d 692 . . . 4  |-  ( x  =  y  ->  (
( AOutsideOf <. x ,  R >.  /\  <. A ,  x >.Cgr
<. B ,  C >. )  <-> 
( AOutsideOf <. y ,  R >.  /\  <. A ,  y
>.Cgr <. B ,  C >. ) ) )
6059reu4 3115 . . 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 646 . 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 424 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 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   E.wrex 2693   E!wreu 2694   <.cop 3804   class class class wbr 4199   ` cfv 5440   NNcn 9984   EEcee 25770    Btwn cbtwn 25771  Cgrccgr 25772  OutsideOfcoutsideof 25996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-sup 7432  df-oi 7463  df-card 7810  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-n0 10206  df-z 10267  df-uz 10473  df-rp 10597  df-ico 10906  df-icc 10907  df-fz 11028  df-fzo 11119  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-sum 12463  df-ee 25773  df-btwn 25774  df-cgr 25775  df-ofs 25860  df-ifs 25916  df-cgr3 25917  df-colinear 25918  df-fs 25919  df-outsideof 25997
  Copyright terms: Public domain W3C validator