Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  btwnxfr Structured version   Unicode version

Theorem btwnxfr 28016
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 28006 . . . . . 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 984 . . . . . 6  |-  ( (
<. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
31, 2syl6bi 228 . . . . 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 983 . . . . . 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 1016 . . . . . 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 1017 . . . . . 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 1018 . . . . . 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 1019 . . . . . 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 1021 . . . . . 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 28015 . . . . . 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 1231 . . . . 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 479 . . . 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 429 . . 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 758 . . . . . . . . . . . . 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 529 . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . 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 1064 . . . . . . . . . . . . . . 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 1066 . . . . . . . . . . . . . . 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 27948 . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . 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 27948 . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 458 . . . . . . . . . . . . . 14  |-  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )
25 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
26 simpl2 987 . . . . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . . 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 28012 . . . . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . . . 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 28013 . . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . . 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 1065 . . . . . . . . . . . . . . . . . 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 28006 . . . . . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . . . . . . . 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 27961 . . . . . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . 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 44 . . . . . . . . . . . . . 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 651 . . . . . . . . . . . . 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 429 . . . . . . . . . . . 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 1163 . . . . . . . . . . 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 434 . . . . . . . . . 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 28003 . . . . . . . . . . . 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 28004 . . . . . . . . . . . 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 235 . . . . . . . . . . 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 1245 . . . . . . . . . 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 44 . . . . . . . . 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 27963 . . . . . . . . . 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 1215 . . . . . . . . 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 44 . . . . . . . 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 429 . . . . . . 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 4311 . . . . . 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 612 . . . . 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 797 . . . 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 2839 . . 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 434 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 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714   <.cop 3880   class class class wbr 4289   ` cfv 5415   NNcn 10318   EEcee 23069    Btwn cbtwn 23070  Cgrccgr 23071    InnerFiveSeg cifs 27995  Cgr3ccgr3 27996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-ee 23072  df-btwn 23073  df-cgr 23074  df-ofs 27943  df-ifs 28000  df-cgr3 28001
This theorem is referenced by:  colinearxfr  28035  brofs2  28037  brifs2  28038  endofsegid  28045  brsegle2  28069
  Copyright terms: Public domain W3C validator