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Theorem segcon2 29960
Description: Generalization of axsegcon 24357. 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 24357, 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 4459 . . . . 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 2968 . 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 996 . . . . 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 997 . . . . . 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 24357 . . . . 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 1228 . . . 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 999 . . . . . . . . 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 1049 . . . . . . . . 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 1001 . . . . . . . . 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 24357 . . . . . . . . 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 1233 . . . . . . . 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 975 . . . . . . . . . . . . 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 1002 . . . . . . . . . . . . . . . 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 1056 . . . . . . . . . . . . . . . . . 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 999 . . . . . . . . . . . . . . . . . . . 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 1049 . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . . . . . . . . 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 29859 . . . . . . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . 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 29870 . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 29957 . . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . . 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 1327 . . . . . . . . . . . . 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 804 . . . . . . . 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 2932 . . . . . . 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 804 . . . 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 2949 . . 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 1022 . . . 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 998 . . . 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 24357 . . . 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 1233 . . 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 2925 . . 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 2772 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   <.cop 4038   class class class wbr 4456   ` cfv 5594   NNcn 10556   EEcee 24318    Btwn cbtwn 24319  Cgrccgr 24320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-ee 24321  df-btwn 24322  df-cgr 24323  df-ofs 29838  df-colinear 29894  df-ifs 29895  df-cgr3 29896  df-fs 29897
This theorem is referenced by:  seglelin  29971  outsideofeu  29986
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