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Theorem segcon2 27983
Description: Generalization of axsegcon 22996. 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 22996, 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 4283 . . . . 5  |-  ( A  =  Q  ->  ( A  Btwn  <. Q ,  x >.  <-> 
Q  Btwn  <. Q ,  x >. ) )
21orbi1d 695 . . . 4  |-  ( A  =  Q  ->  (
( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  <-> 
( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
32anbi1d 697 . . 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 2726 . 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 981 . . . . 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 982 . . . . . 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 449 . . . . 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 22996 . . . . 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 1211 . . . 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 462 . . 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 984 . . . . . . . . 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 458 . . . . . . . . 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 1034 . . . . . . . . 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 986 . . . . . . . . 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 22996 . . . . . . . . 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 1216 . . . . . . . 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 462 . . . . . . 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 642 . . . . . . . . . 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 960 . . . . . . . . . . . . 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 987 . . . . . . . . . . . . . . . 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 1041 . . . . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . . . . . . 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 1034 . . . . . . . . . . . . . . . . . . . 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 748 . . . . . . . . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . . . . . . 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 27882 . . . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 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 2633 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . 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 1040 . . . . . . . . . . . . . . 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 27893 . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . . 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 27980 . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 1310 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 610 . . . . . . . . . . 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 559 . . . . . . . . . 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 469 . . . . . . . . 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 795 . . . . . . . 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 2818 . . . . . . 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 610 . . . . 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 795 . . . 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 2831 . . 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 1007 . . . 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 983 . . . 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 22996 . . . 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 1216 . . 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 385 . . . . 5  |-  ( Q 
Btwn  <. Q ,  x >.  ->  ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) )
5756anim1i 563 . . . 4  |-  ( ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
5857reximi 2813 . . 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 16 . 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 2676 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 184    \/ wo 368    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   <.cop 3871   class class class wbr 4280   ` cfv 5406   NNcn 10310   EEcee 22957    Btwn cbtwn 22958  Cgrccgr 22959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-ee 22960  df-btwn 22961  df-cgr 22962  df-ofs 27861  df-colinear 27917  df-ifs 27918  df-cgr3 27919  df-fs 27920
This theorem is referenced by:  seglelin  27994  outsideofeu  28009
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