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Theorem btwnxfr 24019
Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
btwnxfr  |-  ( ( 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 ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E  Btwn  <. D ,  F >. ) )

Proof of Theorem btwnxfr
StepHypRef Expression
1 brcgr3 24009 . . . . . 6  |-  ( ( 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 >. ) ) )
2 simp2 961 . . . . . 6  |-  ( (
<. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
31, 2syl6bi 221 . . . . 5  |-  ( ( 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 ,  C >.Cgr <. D ,  F >. ) )
4 simp1 960 . . . . . 6  |-  ( ( 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 ) ) )  ->  N  e.  NN )
5 simp21 993 . . . . . 6  |-  ( ( 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  e.  ( EE `  N
) )
6 simp22 994 . . . . . 6  |-  ( ( 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 ) ) )  ->  B  e.  ( EE `  N
) )
7 simp23 995 . . . . . 6  |-  ( ( 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 ) ) )  ->  C  e.  ( EE `  N
) )
8 simp31 996 . . . . . 6  |-  ( ( 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 ) ) )  ->  D  e.  ( EE `  N
) )
9 simp33 998 . . . . . 6  |-  ( ( 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 ) ) )  ->  F  e.  ( EE `  N
) )
10 cgrxfr 24018 . . . . . 6  |-  ( ( 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 >. >. ) ) )
114, 5, 6, 7, 8, 9, 10syl132anc 1205 . . . . 5  |-  ( ( 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 ) ) )  ->  (
( 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 >. >. )
) )
123, 11sylan2d 470 . . . 4  |-  ( ( 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 ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
) )
1312imp 420 . . 3  |-  ( ( ( 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 ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
) )  ->  E. e  e.  ( EE `  N
) ( e  Btwn  <. D ,  F >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
)
14 simprrl 743 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  e  Btwn  <. D ,  F >. )
1514, 14jca 520 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  ( e  Btwn  <. D ,  F >.  /\  e  Btwn  <. D ,  F >. ) )
16 simpl1 963 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  N  e.  NN )
17 simpl31 1041 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  D  e.  ( EE `  N
) )
18 simpl33 1043 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  F  e.  ( EE `  N
) )
1916, 17, 18cgrrflxd 23951 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  <. D ,  F >.Cgr <. D ,  F >. )
20 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  e  e.  ( EE `  N
) )
2116, 20, 18cgrrflxd 23951 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  <. e ,  F >.Cgr <. e ,  F >. )
2219, 21jca 520 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( <. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <.
e ,  F >. ) )
2322adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  ( <. D ,  F >.Cgr <. D ,  F >.  /\  <. e ,  F >.Cgr
<. e ,  F >. ) )
24 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )
25 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
26 simpl2 964 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )
27 simpl3 965 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )
2817, 20, 183jca 1137 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( D  e.  ( EE `  N )  /\  e  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )
29 cgr3tr4 24015 . . . . . . . . . . . . . . . 16  |-  ( ( 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 ) )  /\  ( D  e.  ( EE `  N )  /\  e  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  <. D ,  <. E ,  F >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )
3016, 26, 27, 28, 29syl13anc 1189 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
)  ->  <. D ,  <. E ,  F >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )
31 cgr3com 24016 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  e  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. D ,  <. E ,  F >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  <. D ,  <. e ,  F >. >.Cgr3 <. D ,  <. E ,  F >. >. ) )
3216, 27, 17, 20, 18, 31syl113anc 1199 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( <. D ,  <. E ,  F >. >.Cgr3 <. D ,  <. e ,  F >. >.  <->  <. D ,  <. e ,  F >. >.Cgr3 <. D ,  <. E ,  F >. >. ) )
33 simpl32 1042 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  E  e.  ( EE `  N
) )
34 brcgr3 24009 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. D ,  <. e ,  F >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. D ,  e >.Cgr <. D ,  E >.  /\  <. D ,  F >.Cgr <. D ,  F >.  /\  <. e ,  F >.Cgr
<. E ,  F >. ) ) )
3516, 17, 20, 18, 17, 33, 18, 34syl133anc 1210 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( <. D ,  <. e ,  F >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. D ,  e >.Cgr <. D ,  E >.  /\  <. D ,  F >.Cgr <. D ,  F >.  /\  <. e ,  F >.Cgr
<. E ,  F >. ) ) )
36 simpr1 966 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  ( <. D ,  e >.Cgr <. D ,  E >.  /\ 
<. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <. E ,  F >. ) )  ->  <. D , 
e >.Cgr <. D ,  E >. )
37 simpr3 968 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  ( <. D ,  e >.Cgr <. D ,  E >.  /\ 
<. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <. E ,  F >. ) )  ->  <. e ,  F >.Cgr <. E ,  F >. )
3816, 20, 18, 33, 18, 37cgrcomlrand 23964 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  ( <. D ,  e >.Cgr <. D ,  E >.  /\ 
<. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <. E ,  F >. ) )  ->  <. F , 
e >.Cgr <. F ,  E >. )
3936, 38jca 520 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  ( <. D ,  e >.Cgr <. D ,  E >.  /\ 
<. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <. E ,  F >. ) )  ->  ( <. D ,  e >.Cgr <. D ,  E >.  /\  <. F , 
e >.Cgr <. F ,  E >. ) )
4039ex 425 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( <. D ,  e
>.Cgr <. D ,  E >.  /\  <. D ,  F >.Cgr
<. D ,  F >.  /\ 
<. e ,  F >.Cgr <. E ,  F >. )  ->  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) ) )
4135, 40sylbid 208 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( <. D ,  <. e ,  F >. >.Cgr3 <. D ,  <. E ,  F >. >.  ->  ( <. D ,  e >.Cgr <. D ,  E >.  /\ 
<. F ,  e >.Cgr <. F ,  E >. ) ) )
4232, 41sylbid 208 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( <. D ,  <. E ,  F >. >.Cgr3 <. D ,  <. e ,  F >. >.  ->  ( <. D ,  e >.Cgr <. D ,  E >.  /\ 
<. F ,  e >.Cgr <. F ,  E >. ) ) )
4330, 42syld 42 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >.
)  ->  ( <. D ,  e >.Cgr <. D ,  E >.  /\  <. F , 
e >.Cgr <. F ,  E >. ) ) )
4424, 25, 43syl2ani 640 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )  -> 
( <. D ,  e
>.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) ) )
4544imp 420 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) )
4615, 23, 453jca 1137 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  ( ( e 
Btwn  <. D ,  F >.  /\  e  Btwn  <. D ,  F >. )  /\  ( <. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <.
e ,  F >. )  /\  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) ) )
4746ex 425 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )  -> 
( ( e  Btwn  <. D ,  F >.  /\  e  Btwn  <. D ,  F >. )  /\  ( <. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <.
e ,  F >. )  /\  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) ) ) )
48 brifs 24006 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  D  e.  ( EE
`  N )  /\  e  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  (
e  e.  ( EE
`  N )  /\  F  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( <. <. D ,  e
>. ,  <. F , 
e >. >. 
InnerFiveSeg  <. <. D ,  e
>. ,  <. F ,  E >. >. 
<->  ( ( e  Btwn  <. D ,  F >.  /\  e  Btwn  <. D ,  F >. )  /\  ( <. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <.
e ,  F >. )  /\  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) ) ) )
49 ifscgr 24007 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  D  e.  ( EE
`  N )  /\  e  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  (
e  e.  ( EE
`  N )  /\  F  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( <. <. D ,  e
>. ,  <. F , 
e >. >. 
InnerFiveSeg  <. <. D ,  e
>. ,  <. F ,  E >. >.  ->  <. e ,  e >.Cgr <. e ,  E >. ) )
5048, 49sylbird 228 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  D  e.  ( EE
`  N )  /\  e  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  (
e  e.  ( EE
`  N )  /\  F  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( ( e 
Btwn  <. D ,  F >.  /\  e  Btwn  <. D ,  F >. )  /\  ( <. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <.
e ,  F >. )  /\  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) )  ->  <. e ,  e >.Cgr <. e ,  E >. ) )
5116, 17, 20, 18, 20, 17, 20, 18, 33, 50syl333anc 1219 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( ( e  Btwn  <. D ,  F >.  /\  e  Btwn  <. D ,  F >. )  /\  ( <. D ,  F >.Cgr <. D ,  F >.  /\ 
<. e ,  F >.Cgr <.
e ,  F >. )  /\  ( <. D , 
e >.Cgr <. D ,  E >.  /\  <. F ,  e
>.Cgr <. F ,  E >. ) )  ->  <. e ,  e >.Cgr <. e ,  E >. ) )
5247, 51syld 42 . . . . . . . . 9  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )  ->  <. e ,  e >.Cgr <. e ,  E >. ) )
53 cgrid2 23966 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( e  e.  ( EE `  N )  /\  e  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( <. e ,  e >.Cgr <. e ,  E >.  -> 
e  =  E ) )
5416, 20, 20, 33, 53syl13anc 1189 . . . . . . . . 9  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  ( <. e ,  e >.Cgr <. e ,  E >.  -> 
e  =  E ) )
5552, 54syld 42 . . . . . . . 8  |-  ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) )  -> 
e  =  E ) )
5655imp 420 . . . . . . 7  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  e  =  E )
5756, 14eqbrtrrd 3985 . . . . . 6  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  /\  (
e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. ) ) )  ->  E  Btwn  <. D ,  F >. )
5857expr 601 . . . . 5  |-  ( ( ( ( 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 ) ) )  /\  e  e.  ( EE `  N
) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
) )  ->  (
( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  E  Btwn  <. D ,  F >. ) )
5958an32s 782 . . . 4  |-  ( ( ( ( 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 ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
) )  /\  e  e.  ( EE `  N
) )  ->  (
( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  E  Btwn  <. D ,  F >. ) )
6059rexlimdva 2638 . . 3  |-  ( ( ( 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 ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
) )  ->  ( E. e  e.  ( EE `  N ) ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  E  Btwn  <. D ,  F >. ) )
6113, 60mpd 16 . 2  |-  ( ( ( 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 ) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.
) )  ->  E  Btwn  <. D ,  F >. )
6261ex 425 1  |-  ( ( 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 ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  E  Btwn  <. D ,  F >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2517   <.cop 3584   class class class wbr 3963   ` cfv 4638   NNcn 9679   EEcee 23856    Btwn cbtwn 23857  Cgrccgr 23858    InnerFiveSeg cifs 23998  Cgr3ccgr3 23999
This theorem is referenced by:  colinearxfr  24038  brofs2  24040  brifs2  24041  endofsegid  24048  brsegle2  24072
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-ee 23859  df-btwn 23860  df-cgr 23861  df-ofs 23946  df-ifs 24002  df-cgr3 24003
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