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Theorem cgrxfr 30766
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
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1008 . . . 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 1061 . . . 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 1060 . . . 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 30738 . . . 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 1264 . . 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 1008 . . . . . . . . 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 462 . . . . . . . . 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 1060 . . . . . . . . 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 1083 . . . . . . . . 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 1084 . . . . . . . . 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 24894 . . . . . . . . 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 1273 . . . . . . . 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 466 . . . . . . 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 653 . . . . . . . . . 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 1008 . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . . 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 1085 . . . . . . . . . . . . . 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 24894 . . . . . . . . . . . . . 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 1273 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 653 . . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . . 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 255 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . . . . . 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 1008 . . . . . . . . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . . . . . . . . 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 1060 . . . . . . . . . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . . . 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 30737 . . . . . . . . . . . . . . . . . . . . . 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 1083 . . . . . . . . . . . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . . . . . . . . . . . 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 1085 . . . . . . . . . . . . . . . . . . . . . . 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 30732 . . . . . . . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . . . 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 30720 . . . . . . . . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . . . . . . . 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 1061 . . . . . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . 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 30726 . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . 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 30702 . . . . . . . . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . . . . 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 30721 . . . . . . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . . . . . . 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 45 . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . . 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 4126 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. g ,  f >.  =  <. g ,  F >. )
6564breq2d 4373 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  (
e  Btwn  <. g ,  f >.  <->  e  Btwn  <. g ,  F >. ) )
66 opeq2 4126 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. e ,  f >.  =  <. e ,  F >. )
6766breq1d 4371 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  ( <. e ,  f >.Cgr <. B ,  C >.  <->  <. e ,  F >.Cgr <. B ,  C >. ) )
6865, 67anbi12d 715 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  =  F  ->  (
( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. )  <->  ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr
<. B ,  C >. ) ) )
6968biimpa 486 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  F  /\  ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )
70 simpl 458 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. )  ->  e  Btwn  <.
g ,  F >. )
71 btwnexch3 30731 . . . . . . . . . . . . . . . . . . . . . . . . 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 1273 . . . . . . . . . . . . . . . . . . . . . . . 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 660 . . . . . . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . . . . 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 30702 . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . . . . . 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 30702 . . . . . . . . . . . . . . . . . . . . . . 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 30757 . . . . . . . . . . . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . 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 1188 . . . . . . . . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . . . . . . 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 >. >. ) ) )
8887expcomd 439 . . . . . . . . . . . . . . . . . 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 623 . . . . . . . . . . . . . . . . 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 15 . . . . . . . . . . . . . . . 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 618 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . 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 2851 . . . . . . . . . . . 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 15 . . . . . . . . . . 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 618 . . . . . . . . . 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 474 . . . . . . . . 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 811 . . . . . . . 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 2834 . . . . . . 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 15 . . . . . 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 618 . . . . 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 811 . . . 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 2851 . . 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 15 . 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 435 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   E.wrex 2710   <.cop 3942   class class class wbr 4361   ` cfv 5539   NNcn 10555   EEcee 24855    Btwn cbtwn 24856  Cgrccgr 24857  Cgr3ccgr3 30747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-sup 7904  df-oi 7973  df-card 8320  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-ico 11587  df-icc 11588  df-fz 11731  df-fzo 11862  df-seq 12159  df-exp 12218  df-hash 12461  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-clim 13490  df-sum 13691  df-ee 24858  df-btwn 24859  df-cgr 24860  df-ofs 30694  df-cgr3 30752
This theorem is referenced by:  btwnxfr  30767  lineext  30787  seglecgr12im  30821  segletr  30825
  Copyright terms: Public domain W3C validator