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Theorem cgrxfr 29623
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 999 . . . 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 1052 . . . 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 1051 . . . 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 29595 . . . 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 1228 . . 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 999 . . . . . . . . 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 461 . . . . . . . . 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 1051 . . . . . . . . 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 1074 . . . . . . . . 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 1075 . . . . . . . . 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 24044 . . . . . . . . 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 1237 . . . . . . . 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 465 . . . . . . 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 649 . . . . . . . . . 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 999 . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . 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 1076 . . . . . . . . . . . . . 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 24044 . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 649 . . . . . . . . . . . . . . . 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 975 . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . 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 252 . . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . . . . 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 999 . . . . . . . . . . . . . . . . . . . . . . 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 1002 . . . . . . . . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . . . . . . . . . . . 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 1004 . . . . . . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . . . 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 29594 . . . . . . . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . . . . . . . . . . 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 1076 . . . . . . . . . . . . . . . . . . . . . . 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 29589 . . . . . . . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 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 29577 . . . . . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . . . . . 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 1052 . . . . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 29583 . . . . . . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 29559 . . . . . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . . 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 29578 . . . . . . . . . . . . . . . . . . . 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 1251 . . . . . . . . . . . . . . . . . . 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 44 . . . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . . . . 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 4220 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. g ,  f >.  =  <. g ,  F >. )
6564breq2d 4465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  (
e  Btwn  <. g ,  f >.  <->  e  Btwn  <. g ,  F >. ) )
66 opeq2 4220 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( f  =  F  ->  <. e ,  f >.  =  <. e ,  F >. )
6766breq1d 4463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f  =  F  ->  ( <. e ,  f >.Cgr <. B ,  C >.  <->  <. e ,  F >.Cgr <. B ,  C >. ) )
6865, 67anbi12d 710 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f  =  F  ->  (
( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. )  <->  ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr
<. B ,  C >. ) ) )
6968biimpa 484 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f  =  F  /\  ( e  Btwn  <. g ,  f >.  /\  <. e ,  f >.Cgr <. B ,  C >. ) )  -> 
( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. ) )
70 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( e  Btwn  <. g ,  F >.  /\  <. e ,  F >.Cgr <. B ,  C >. )  ->  e  Btwn  <.
g ,  F >. )
71 btwnexch3 29588 . . . . . . . . . . . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . . . 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 29559 . . . . . . . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . . 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 29559 . . . . . . . . . . . . . . . . . . . . . . 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 29614 . . . . . . . . . . . . . . . . . . . . . . . . 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 1251 . . . . . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . . . . 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 1179 . . . . . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . . . . . . 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 438 . . . . . . . . . . . . . . . . . 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 619 . . . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . 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 2959 . . . . . . . . . . . 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 615 . . . . . . . . . 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 472 . . . . . . . . 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 802 . . . . . . . 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 2942 . . . . . . 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 615 . . . . 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 802 . . . 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 2959 . . 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 434 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   <.cop 4039   class class class wbr 4453   ` cfv 5594   NNcn 10548   EEcee 24005    Btwn cbtwn 24006  Cgrccgr 24007  Cgr3ccgr3 29604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-ee 24008  df-btwn 24009  df-cgr 24010  df-ofs 29551  df-cgr3 29609
This theorem is referenced by:  btwnxfr  29624  lineext  29644  seglecgr12im  29678  segletr  29682
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