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Theorem segcon2 26040
Description: Generalization of axsegcon 25867. 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 25867, 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 4216 . . . . 5  |-  ( A  =  Q  ->  ( A  Btwn  <. Q ,  x >.  <-> 
Q  Btwn  <. Q ,  x >. ) )
21orbi1d 685 . . . 4  |-  ( A  =  Q  ->  (
( A  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  <-> 
( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) ) )
32anbi1d 687 . . 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 2727 . 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 958 . . . . 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 959 . . . . . 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 440 . . . . 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 25867 . . . . 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 1185 . . . 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 453 . . 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 961 . . . . . . . . 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 449 . . . . . . . . 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 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 ) )  ->  Q  e.  ( EE `  N ) )
14 simpl3 963 . . . . . . . . 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 25867 . . . . . . . . 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 1190 . . . . . . . 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 453 . . . . . . 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 632 . . . . . . . . . 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 939 . . . . . . . . . . . . 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 964 . . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . . . . . 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 961 . . . . . . . . . . . . . . . . . . . 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 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 ) ) )  ->  Q  e.  ( EE `  N
) )
24 simprl 734 . . . . . . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . . . . 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 25939 . . . . . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . 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 2637 . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . 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 2688 . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . 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 25950 . . . . . . . . . . . . . 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 966 . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 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 26037 . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 1283 . . . . . . . . . . . . 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 464 . . . . . . . . . . . 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 600 . . . . . . . . . . 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 549 . . . . . . . . . 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 460 . . . . . . . . 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 781 . . . . . . . 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 2819 . . . . . . 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 600 . . . . 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 781 . . . 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 984 . . . 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 960 . . . 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 25867 . . . 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 1190 . . 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 376 . . . . 5  |-  ( Q 
Btwn  <. Q ,  x >.  ->  ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. ) )
5756anim1i 553 . . . 4  |-  ( ( Q  Btwn  <. Q ,  x >.  /\  <. Q ,  x >.Cgr <. B ,  C >. )  ->  ( ( Q  Btwn  <. Q ,  x >.  \/  x  Btwn  <. Q ,  A >. )  /\  <. Q ,  x >.Cgr <. B ,  C >. ) )
5857reximi 2814 . . 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 2680 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 set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   <.cop 3818   class class class wbr 4213   ` cfv 5455   NNcn 10001   EEcee 25828    Btwn cbtwn 25829  Cgrccgr 25830
This theorem is referenced by:  seglelin  26051  outsideofeu  26066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-sup 7447  df-oi 7480  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-sum 12481  df-ee 25831  df-btwn 25832  df-cgr 25833  df-ofs 25918  df-ifs 25974  df-cgr3 25975  df-colinear 25976  df-fs 25977
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