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Theorem cgrxfr 24018
Description: A line segment can be divided at the same place as a congruent line segment is divided. Theorem 4.5 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
cgrxfr  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
Distinct variable groups:    A, e    B, e    C, e    D, e   
e, F    e, N

Proof of Theorem cgrxfr
StepHypRef Expression
1 simpl1 963 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  N  e.  NN )
2 simpl3r 1016 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  F  e.  ( EE `  N
) )
3 simpl3l 1015 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  D  e.  ( EE `  N
) )
4 btwndiff 23990 . . . 4  |-  ( ( N  e.  NN  /\  F  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  ->  E. g  e.  ( EE `  N
) ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )
51, 2, 3, 4syl3anc 1187 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  E. g  e.  ( EE `  N
) ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )
6 simpl1 963 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  N  e.  NN )
7 simpr 449 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  -> 
g  e.  ( EE
`  N ) )
8 simpl3l 1015 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  D  e.  ( EE `  N ) )
9 simpl21 1038 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
10 simpl22 1039 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
11 axsegcon 23895 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( g  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  ->  E. e  e.  ( EE `  N ) ( D  Btwn  <. g ,  e >.  /\  <. D , 
e >.Cgr <. A ,  B >. ) )
126, 7, 8, 9, 10, 11syl122anc 1196 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  ->  E. e  e.  ( EE `  N ) ( D  Btwn  <. g ,  e >.  /\  <. D , 
e >.Cgr <. A ,  B >. ) )
1312adantr 453 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  E. e  e.  ( EE `  N
) ( D  Btwn  <.
g ,  e >.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )
14 anass 633 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  e  e.  ( EE `  N ) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) ) )
15 simpl1 963 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  N  e.  NN )
16 simprl 735 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  g  e.  ( EE `  N
) )
17 simprr 736 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  e  e.  ( EE `  N
) )
18 simpl22 1039 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
19 simpl23 1040 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
20 axsegcon 23895 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  E. f  e.  ( EE `  N ) ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )
2115, 16, 17, 18, 19, 20syl122anc 1196 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  ->  E. f  e.  ( EE `  N
) ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. ) )
2221adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  ->  E. f  e.  ( EE `  N ) ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )
23 anass 633 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  f  e.  ( EE `  N
) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( ( g  e.  ( EE `  N
)  /\  e  e.  ( EE `  N ) )  /\  f  e.  ( EE `  N
) ) ) )
24 df-3an 941 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  ( EE
`  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N
) )  <->  ( (
g  e.  ( EE
`  N )  /\  e  e.  ( EE `  N ) )  /\  f  e.  ( EE `  N ) ) )
2524anbi2i 678 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( ( g  e.  ( EE `  N
)  /\  e  e.  ( EE `  N ) )  /\  f  e.  ( EE `  N
) ) ) )
2623, 25bitr4i 245 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  f  e.  ( EE `  N
) )  <->  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) ) )
27 simplrr 740 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  D  =/=  g )
2827ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  =/=  g
)
2928necomd 2502 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  g  =/=  D
)
30 simpl1 963 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  N  e.  NN )
31 simpr1 966 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  g  e.  ( EE `  N
) )
32 simpl3l 1015 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
33 simpr2 967 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  e  e.  ( EE `  N
) )
34 simpr3 968 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  f  e.  ( EE `  N
) )
35 simprl 735 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  D  Btwn  <. g ,  e
>. )
3635ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. g ,  e >. )
37 simprrl 743 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  e  Btwn  <. g ,  f >. )
3830, 31, 32, 33, 34, 36, 37btwnexchand 23989 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. g ,  f >. )
39 simpl21 1038 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
40 simpl22 1039 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
41 simpl23 1040 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
4230, 31, 32, 33, 34, 36, 37btwnexch3and 23984 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  e  Btwn  <. D , 
f >. )
43 simplll 737 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  B  Btwn  <. A ,  C >. )
4443ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  B  Btwn  <. A ,  C >. )
45 simprr 736 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  <. D , 
e >.Cgr <. A ,  B >. )
4645ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  e
>.Cgr <. A ,  B >. )
47 simprrr 744 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. e ,  f
>.Cgr <. B ,  C >. )
4830, 32, 33, 34, 39, 40, 41, 42, 44, 46, 47cgrextendand 23972 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  f
>.Cgr <. A ,  C >. )
4938, 48jca 520 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( D  Btwn  <.
g ,  f >.  /\  <. D ,  f
>.Cgr <. A ,  C >. ) )
50 simpl3r 1016 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
51 simplrl 739 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  D  Btwn  <. F ,  g
>. )
5251ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. F , 
g >. )
5330, 32, 50, 31, 52btwncomand 23978 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  D  Btwn  <. g ,  F >. )
54 simpllr 738 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. )
5554ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
5630, 39, 41, 32, 50, 55cgrcomand 23954 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  F >.Cgr
<. A ,  C >. )
5753, 56jca 520 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( D  Btwn  <.
g ,  F >.  /\ 
<. D ,  F >.Cgr <. A ,  C >. ) )
5829, 49, 573jca 1137 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( g  =/= 
D  /\  ( D  Btwn  <. g ,  f
>.  /\  <. D ,  f
>.Cgr <. A ,  C >. )  /\  ( D 
Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) ) )
5958ex 425 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) )  /\  ( D  Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
( g  =/=  D  /\  ( D  Btwn  <. g ,  f >.  /\  <. D ,  f >.Cgr <. A ,  C >. )  /\  ( D  Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) ) ) )
60 segconeq 23973 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  (
g  e.  ( EE
`  N )  /\  f  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( g  =/= 
D  /\  ( D  Btwn  <. g ,  f
>.  /\  <. D ,  f
>.Cgr <. A ,  C >. )  /\  ( D 
Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) )  ->  f  =  F ) )
6130, 32, 39, 41, 31, 34, 50, 60syl133anc 1210 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( g  =/=  D  /\  ( D  Btwn  <. g ,  f >.  /\  <. D ,  f >.Cgr <. A ,  C >. )  /\  ( D  Btwn  <. g ,  F >.  /\  <. D ,  F >.Cgr
<. A ,  C >. ) )  ->  f  =  F ) )
6259, 61syld 42 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) )  /\  ( D  Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
f  =  F ) )
6362imp 420 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  f  =  F )
64 opeq2 3738 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. g ,  f >.  =  <. g ,  F >. )
6564breq2d 3975 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  (
e  Btwn  <. g ,  f >.  <->  e  Btwn  <. g ,  F >. ) )
66 opeq2 3738 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. e ,  f >.  =  <. e ,  F >. )
6766breq1d 3973 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  ( <. e ,  f >.Cgr <. B ,  C >.  <->  <. e ,  F >.Cgr <. B ,  C >. ) )
6865, 67anbi12d 694 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  =  F  ->  (
( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. )  <->  ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr
<. B ,  C >. ) ) )
6968biimpa 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  F  /\  ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )
70 simpl 445 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. )  ->  e  Btwn  <.
g ,  F >. )
71 btwnexch3 23983 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN  /\  ( g  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( e  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( D  Btwn  <.
g ,  e >.  /\  e  Btwn  <. g ,  F >. )  ->  e  Btwn  <. D ,  F >. ) )
7230, 31, 32, 33, 50, 71syl122anc 1196 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( D  Btwn  <. g ,  e >.  /\  e  Btwn  <. g ,  F >. )  ->  e  Btwn  <. D ,  F >. ) )
7335, 70, 72syl2ani 640 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) )  /\  ( D  Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )  ->  e  Btwn  <. D ,  F >. ) )
7473imp 420 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  -> 
e  Btwn  <. D ,  F >. )
75 simplrr 740 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )  ->  <. D , 
e >.Cgr <. A ,  B >. )
7675adantl 454 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. D ,  e >.Cgr <. A ,  B >. )
7730, 32, 33, 39, 40, 76cgrcomand 23954 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  B >.Cgr <. D ,  e >. )
7854ad2antrl 711 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  C >.Cgr <. D ,  F >. )
79 simprrr 744 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. e ,  F >.Cgr <. B ,  C >. )
8030, 33, 50, 40, 41, 79cgrcomand 23954 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. B ,  C >.Cgr <.
e ,  F >. )
81 brcgr3 24009 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  e  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D , 
e >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. e ,  F >. ) ) )
8230, 39, 40, 41, 32, 33, 50, 81syl133anc 1210 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D , 
e >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. e ,  F >. ) ) )
8382adantr 453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  -> 
( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  (
<. A ,  B >.Cgr <. D ,  e >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <.
e ,  F >. ) ) )
8477, 78, 80, 83mpbir3and 1140 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
8574, 84jca 520 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) ) )  -> 
( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )
8685expr 601 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  F >.  /\ 
<. e ,  F >.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
8769, 86syl5 30 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( f  =  F  /\  ( e 
Btwn  <. g ,  f
>.  /\  <. e ,  f
>.Cgr <. B ,  C >. ) )  ->  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
8887exp3acom23 1368 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( f  =  F  ->  ( e 
Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) ) )
8988impr 605 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( f  =  F  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9063, 89mpd 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr
<. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) )  /\  (
e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) ) )  ->  ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
9190expr 601 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9226, 91sylanb 460 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  f  e.  ( EE `  N
) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9392an32s 782 . . . . . . . . . . . . 13  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  /\  f  e.  ( EE `  N ) )  -> 
( ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9493rexlimdva 2638 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( E. f  e.  ( EE `  N
) ( e  Btwn  <.
g ,  f >.  /\  <. e ,  f
>.Cgr <. B ,  C >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9522, 94mpd 16 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) )  /\  ( D 
Btwn  <. g ,  e
>.  /\  <. D ,  e
>.Cgr <. A ,  B >. ) ) )  -> 
( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )
9695expr 601 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( g  e.  ( EE `  N )  /\  e  e.  ( EE `  N ) ) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. )  /\  ( D 
Btwn  <. F ,  g
>.  /\  D  =/=  g
) ) )  -> 
( ( D  Btwn  <.
g ,  e >.  /\  <. D ,  e
>.Cgr <. A ,  B >. )  ->  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
) ) )
9714, 96sylanb 460 . . . . . . . . 9  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  e  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  (
( D  Btwn  <. g ,  e >.  /\  <. D ,  e >.Cgr <. A ,  B >. )  ->  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
9897an32s 782 . . . . . . . 8  |-  ( ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( D  Btwn  <. g ,  e >.  /\  <. D ,  e >.Cgr <. A ,  B >. )  ->  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
9998reximdva 2626 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  ( E. e  e.  ( EE `  N ) ( D  Btwn  <. g ,  e >.  /\  <. D , 
e >.Cgr <. A ,  B >. )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
10013, 99mpd 16 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  /\  ( D  Btwn  <. F ,  g >.  /\  D  =/=  g ) ) )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
101100expr 601 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  g  e.  ( EE `  N ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  (
( D  Btwn  <. F , 
g >.  /\  D  =/=  g )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
102101an32s 782 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  /\  g  e.  ( EE `  N
) )  ->  (
( D  Btwn  <. F , 
g >.  /\  D  =/=  g )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
103102rexlimdva 2638 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  ( E. g  e.  ( EE `  N ) ( D  Btwn  <. F , 
g >.  /\  D  =/=  g )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
1045, 103mpd 16 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. D ,  F >. ) )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
105104ex 425 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. D ,  F >. )  ->  E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857  Cgrccgr 23858  Cgr3ccgr3 23999
This theorem is referenced by:  btwnxfr  24019  lineext  24039  seglecgr12im  24073  segletr  24077
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-ofs 23946  df-cgr3 24003
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