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Theorem btwnxfr 30616
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 30606 . . . . . 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 1006 . . . . . 6  |-  ( (
<. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
31, 2syl6bi 231 . . . . 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 1005 . . . . . 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 1038 . . . . . 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 1039 . . . . . 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 1040 . . . . . 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 1041 . . . . . 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 1043 . . . . . 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 30615 . . . . . 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 1282 . . . . 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 484 . . . 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 430 . . 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 772 . . . . . . . . . . . . 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 534 . . . . . . . . . . . 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 1008 . . . . . . . . . . . . . . 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 1086 . . . . . . . . . . . . . . 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 1088 . . . . . . . . . . . . . . 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 30548 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 30548 . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 14  |-  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )
25 simpr 462 . . . . . . . . . . . . . 14  |-  ( ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
26 simpl2 1009 . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . 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 30612 . . . . . . . . . . . . . . . 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 1266 . . . . . . . . . . . . . . 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 30613 . . . . . . . . . . . . . . . . 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 1276 . . . . . . . . . . . . . . . 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 1087 . . . . . . . . . . . . . . . . . 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 30606 . . . . . . . . . . . . . . . . . 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 1287 . . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . . . . . . 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 30561 . . . . . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . 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 45 . . . . . . . . . . . . . 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 660 . . . . . . . . . . . . 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 430 . . . . . . . . . . . 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 1185 . . . . . . . . . . 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 435 . . . . . . . . . 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 30603 . . . . . . . . . . . 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 30604 . . . . . . . . . . . 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 238 . . . . . . . . . . 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 1296 . . . . . . . . . 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 45 . . . . . . . . 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 30563 . . . . . . . . . 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 1266 . . . . . . . . 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 45 . . . . . . . 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 430 . . . . . . 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 4448 . . . . . 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 618 . . . . 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 811 . . . 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 2924 . . 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 15 . 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 435 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783   <.cop 4008   class class class wbr 4426   ` cfv 5601   NNcn 10609   EEcee 24772    Btwn cbtwn 24773  Cgrccgr 24774    InnerFiveSeg cifs 30595  Cgr3ccgr3 30596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-ee 24775  df-btwn 24776  df-cgr 24777  df-ofs 30543  df-ifs 30600  df-cgr3 30601
This theorem is referenced by:  colinearxfr  30635  brofs2  30637  brifs2  30638  endofsegid  30645  brsegle2  30669
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