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Theorem segcon2 30872
Description: Generalization of axsegcon 24957. This time, we generate an endpoint for a segment on the ray  Q A congruent to  B C and starting at  Q, as opposed to axsegcon 24957, where the segment starts at  A (Contributed by Scott Fenton, 14-Oct-2013.) (Removed unneeded inequality, 15-Oct-2013.)
Assertion
Ref Expression
segcon2  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
Distinct variable groups:    x, Q    x, N    x, A    x, B    x, C

Proof of Theorem segcon2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 breq1 4405 . . . . 5  |-  ( A  =  Q  ->  ( A  Btwn  <. Q ,  x >.  <-> 
Q  Btwn  <. Q ,  x >. ) )
21orbi1d 709 . . . 4  |-  ( A  =  Q  ->  (
( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  <-> 
( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
32anbi1d 711 . . 3  |-  ( A  =  Q  ->  (
( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. )  <->  ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
43rexbidv 2901 . 2  |-  ( A  =  Q  ->  ( E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr
<. B ,  C >. )  <->  E. x  e.  ( EE `  N ) ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr
<. B ,  C >. ) ) )
5 simp1 1008 . . . . 5  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  N  e.  NN )
6 simp2 1009 . . . . . 6  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( Q  e.  ( EE `  N
)  /\  A  e.  ( EE `  N ) ) )
76ancomd 453 . . . . 5  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( A  e.  ( EE `  N
)  /\  Q  e.  ( EE `  N ) ) )
8 axsegcon 24957 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) ) )  ->  E. a  e.  ( EE `  N ) ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) )
95, 7, 7, 8syl3anc 1268 . . . 4  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. a  e.  ( EE `  N ) ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) )
109adantr 467 . . 3  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  A  =/=  Q )  ->  E. a  e.  ( EE `  N ) ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) )
11 simpl1 1011 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  ->  N  e.  NN )
12 simpr 463 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  -> 
a  e.  ( EE
`  N ) )
13 simpl2l 1061 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  ->  Q  e.  ( EE `  N ) )
14 simpl3 1013 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  -> 
( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
15 axsegcon 24957 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( a  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  E. x  e.  ( EE `  N ) ( Q  Btwn  <. a ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
1611, 12, 13, 14, 15syl121anc 1273 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  ->  E. x  e.  ( EE `  N ) ( Q  Btwn  <. a ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
1716adantr 467 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) ) )  ->  E. x  e.  ( EE `  N ) ( Q  Btwn  <. a ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
18 anass 655 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  x  e.  ( EE `  N ) )  <->  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) ) )
19 df-3an 987 . . . . . . . . . . . . 13  |-  ( ( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. )  <-> 
( ( A  =/= 
Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. ) )  /\  Q  Btwn  <. a ,  x >. ) )
20 simpr1 1014 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  A  =/=  Q )
21 simpr2r 1068 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  <. Q , 
a >.Cgr <. A ,  Q >. )
22 simpl1 1011 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  N  e.  NN )
23 simpl2l 1061 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  Q  e.  ( EE `  N
) )
24 simprl 764 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  a  e.  ( EE `  N
) )
25 simpl2r 1062 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
26 cgrdegen 30771 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  a  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) ) )  ->  ( <. Q , 
a >.Cgr <. A ,  Q >.  ->  ( Q  =  a  <->  A  =  Q
) ) )
2722, 23, 24, 25, 23, 26syl122anc 1277 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  ( <. Q ,  a >.Cgr <. A ,  Q >.  -> 
( Q  =  a  <-> 
A  =  Q ) ) )
2827adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  ( <. Q ,  a >.Cgr <. A ,  Q >.  ->  ( Q  =  a  <->  A  =  Q
) ) )
2921, 28mpd 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  ( Q  =  a  <->  A  =  Q
) )
3029necon3bid 2668 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  ( Q  =/=  a  <->  A  =/=  Q
) )
3120, 30mpbird 236 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  Q  =/=  a )
3231necomd 2679 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  a  =/=  Q )
33 simpr2l 1067 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  Q  Btwn  <. A ,  a >. )
3422, 23, 25, 24, 33btwncomand 30782 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  Q  Btwn  <.
a ,  A >. )
35 simpr3 1016 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  Q  Btwn  <.
a ,  x >. )
36 simprr 766 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  x  e.  ( EE `  N
) )
37 btwnconn2 30869 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  ( a  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  x  e.  ( EE `  N
) ) )  -> 
( ( a  =/= 
Q  /\  Q  Btwn  <.
a ,  A >.  /\  Q  Btwn  <. a ,  x >. )  ->  ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
3822, 24, 23, 25, 36, 37syl122anc 1277 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  ->  (
( a  =/=  Q  /\  Q  Btwn  <. a ,  A >.  /\  Q  Btwn  <.
a ,  x >. )  ->  ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
3938adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  ( (
a  =/=  Q  /\  Q  Btwn  <. a ,  A >.  /\  Q  Btwn  <. a ,  x >. )  ->  ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
4032, 34, 35, 39mp3and 1367 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. )  /\  Q  Btwn  <.
a ,  x >. ) )  ->  ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) )
4119, 40sylan2br 479 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  (
( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) )  /\  Q  Btwn  <. a ,  x >. ) )  ->  ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) )
4241expr 620 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. ) ) )  -> 
( Q  Btwn  <. a ,  x >.  ->  ( A 
Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
4342anim1d 568 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  ( a  e.  ( EE `  N )  /\  x  e.  ( EE `  N ) ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A ,  a
>.  /\  <. Q ,  a
>.Cgr <. A ,  Q >. ) ) )  -> 
( ( Q  Btwn  <.
a ,  x >.  /\ 
<. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( A 
Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
4418, 43sylanb 475 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  x  e.  ( EE `  N ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) ) )  -> 
( ( Q  Btwn  <.
a ,  x >.  /\ 
<. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( A 
Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
4544an32s 813 . . . . . . . 8  |-  ( ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) ) )  /\  x  e.  ( EE `  N ) )  -> 
( ( Q  Btwn  <.
a ,  x >.  /\ 
<. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( A 
Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
4645reximdva 2862 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) ) )  -> 
( E. x  e.  ( EE `  N
) ( Q  Btwn  <.
a ,  x >.  /\ 
<. Q ,  x >.Cgr <. B ,  C >. )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
4717, 46mpd 15 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  ( A  =/=  Q  /\  ( Q  Btwn  <. A , 
a >.  /\  <. Q , 
a >.Cgr <. A ,  Q >. ) ) )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr
<. B ,  C >. ) )
4847expr 620 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  a  e.  ( EE `  N ) )  /\  A  =/=  Q )  -> 
( ( Q  Btwn  <. A ,  a >.  /\ 
<. Q ,  a >.Cgr <. A ,  Q >. )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
4948an32s 813 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  A  =/=  Q )  /\  a  e.  ( EE `  N ) )  -> 
( ( Q  Btwn  <. A ,  a >.  /\ 
<. Q ,  a >.Cgr <. A ,  Q >. )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
5049rexlimdva 2879 . . 3  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  A  =/=  Q )  -> 
( E. a  e.  ( EE `  N
) ( Q  Btwn  <. A ,  a >.  /\ 
<. Q ,  a >.Cgr <. A ,  Q >. )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) ) )
5110, 50mpd 15 . 2  |-  ( ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  /\  A  =/=  Q )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr
<. B ,  C >. ) )
52 simp2l 1034 . . . 4  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  Q  e.  ( EE `  N ) )
53 simp3 1010 . . . 4  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )
54 axsegcon 24957 . . . 4  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  Q  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
555, 52, 52, 53, 54syl121anc 1273 . . 3  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
56 orc 387 . . . . 5  |-  ( Q 
Btwn  <. Q ,  x >.  ->  ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) )
5756anim1i 572 . . . 4  |-  ( ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
5857reximi 2855 . . 3  |-  ( E. x  e.  ( EE
`  N ) ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. )  ->  E. x  e.  ( EE `  N
) ( ( Q 
Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
5955, 58syl 17 . 2  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
604, 51, 59pm2.61ne 2709 1  |-  ( ( N  e.  NN  /\  ( Q  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. x  e.  ( EE `  N ) ( ( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   <.cop 3974   class class class wbr 4402   ` cfv 5582   NNcn 10609   EEcee 24918    Btwn cbtwn 24919  Cgrccgr 24920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-ee 24921  df-btwn 24922  df-cgr 24923  df-ofs 30750  df-colinear 30806  df-ifs 30807  df-cgr3 30808  df-fs 30809
This theorem is referenced by:  seglelin  30883  outsideofeu  30898
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