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Theorem btwnxfr 24090
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
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 brcgr3 24080 . . . . . 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 956 . . . . . 6  |-  ( (
<. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
31, 2syl6bi 219 . . . . 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 955 . . . . . 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 988 . . . . . 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 989 . . . . . 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 990 . . . . . 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 991 . . . . . 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 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 ) ) )  ->  F  e.  ( EE `  N
) )
10 cgrxfr 24089 . . . . . 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 1200 . . . . 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 468 . . . 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 418 . . 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 740 . . . . . . . . . . . . 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 518 . . . . . . . . . . . 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 958 . . . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . . 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 1038 . . . . . . . . . . . . . . 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 24022 . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . 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 24022 . . . . . . . . . . . . . 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 518 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 447 . . . . . . . . . . . . . 14  |-  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )
25 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
26 simpl2 959 . . . . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . 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 24086 . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . 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 24087 . . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . . . . . . . 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 24080 . . . . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . . . 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 961 . . . . . . . . . . . . . . . . . . 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 963 . . . . . . . . . . . . . . . . . . . 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 24035 . . . . . . . . . . . . . . . . . . 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 518 . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . 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 40 . . . . . . . . . . . . . 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 637 . . . . . . . . . . . . 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 418 . . . . . . . . . . . 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 1132 . . . . . . . . . . 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 423 . . . . . . . . . 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 24077 . . . . . . . . . . . 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 24078 . . . . . . . . . . . 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 226 . . . . . . . . . . 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 1214 . . . . . . . . . 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 40 . . . . . . . . 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 24037 . . . . . . . . . 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 1184 . . . . . . . . 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 40 . . . . . . . 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 418 . . . . . . 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 4045 . . . . . 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 598 . . . . 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 779 . . . 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 2667 . . 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 14 . 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 423 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 23927    Btwn cbtwn 23928  Cgrccgr 23929    InnerFiveSeg cifs 24069  Cgr3ccgr3 24070
This theorem is referenced by:  colinearxfr  24109  brofs2  24111  brifs2  24112  endofsegid  24119  brsegle2  24143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 23930  df-btwn 23931  df-cgr 23932  df-ofs 24017  df-ifs 24073  df-cgr3 24074
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