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Theorem segcon2 28303
Description: Generalization of axsegcon 23352. 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 23352, 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 4406 . . . . 5  |-  ( A  =  Q  ->  ( A  Btwn  <. Q ,  x >.  <-> 
Q  Btwn  <. Q ,  x >. ) )
21orbi1d 702 . . . 4  |-  ( A  =  Q  ->  (
( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  <-> 
( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
32anbi1d 704 . . 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 2868 . 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 988 . . . . 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 989 . . . . . 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 451 . . . . 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 23352 . . . . 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 1219 . . . 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 465 . . 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 991 . . . . . . . . 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 461 . . . . . . . . 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 1041 . . . . . . . . 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 993 . . . . . . . . 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 23352 . . . . . . . . 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 1224 . . . . . . . 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 465 . . . . . . 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 649 . . . . . . . . . 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 967 . . . . . . . . . . . . 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 994 . . . . . . . . . . . . . . . 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 1048 . . . . . . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . . . . . . . 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 1041 . . . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . . . 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 1042 . . . . . . . . . . . . . . . . . . . 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 28202 . . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 2710 . . . . . . . . . . . . . . . 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 2723 . . . . . . . . . . . . . 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 1047 . . . . . . . . . . . . . . 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 28213 . . . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . 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 28300 . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . 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 476 . . . . . . . . . . . 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 615 . . . . . . . . . . 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 564 . . . . . . . . . 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 472 . . . . . . . . 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 802 . . . . . . . 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 2934 . . . . . . 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 615 . . . . 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 802 . . . 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 2947 . . 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 1014 . . . 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 990 . . . 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 23352 . . . 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 1224 . . 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 568 . . . 4  |-  ( ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
5857reximi 2929 . . 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 2767 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   <.cop 3994   class class class wbr 4403   ` cfv 5529   NNcn 10437   EEcee 23313    Btwn cbtwn 23314  Cgrccgr 23315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-sum 13286  df-ee 23316  df-btwn 23317  df-cgr 23318  df-ofs 28181  df-colinear 28237  df-ifs 28238  df-cgr3 28239  df-fs 28240
This theorem is referenced by:  seglelin  28314  outsideofeu  28329
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