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

Theorem btwnxfr 30872
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 30862 . . . . . 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 1015 . . . . . 6  |-  ( (
<. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  ->  <. A ,  C >.Cgr
<. D ,  F >. )
31, 2syl6bi 236 . . . . 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 1014 . . . . . 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 1047 . . . . . 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 1048 . . . . . 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 1049 . . . . . 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 1050 . . . . . 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 1052 . . . . . 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 30871 . . . . . 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 1294 . . . . 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 489 . . . 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 435 . . 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 779 . . . . . . . . . . . . 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 539 . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . 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 1095 . . . . . . . . . . . . . . 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 1097 . . . . . . . . . . . . . . 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 30804 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . 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 30804 . . . . . . . . . . . . . 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 539 . . . . . . . . . . . . 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 471 . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 14  |-  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >. )
25 simpr 467 . . . . . . . . . . . . . 14  |-  ( ( e  Btwn  <. D ,  F >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. e ,  F >. >. )
26 simpl2 1018 . . . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . . . . 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 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.  ( EE `  N )  /\  e  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )
29 cgr3tr4 30868 . . . . . . . . . . . . . . . 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 1278 . . . . . . . . . . . . . . 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 30869 . . . . . . . . . . . . . . . . 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 1288 . . . . . . . . . . . . . . . 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 1096 . . . . . . . . . . . . . . . . . 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 30862 . . . . . . . . . . . . . . . . . 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 1299 . . . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . . . . . . . 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 30817 . . . . . . . . . . . . . . . . . . 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 539 . . . . . . . . . . . . . . . . . 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 440 . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . 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 435 . . . . . . . . . . . 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 1194 . . . . . . . . . . 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 440 . . . . . . . . . 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 30859 . . . . . . . . . . . 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 30860 . . . . . . . . . . . 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 243 . . . . . . . . . . 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 1308 . . . . . . . . . 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 30819 . . . . . . . . . 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 1278 . . . . . . . . 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 435 . . . . . . 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 4439 . . . . . 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 624 . . . . 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 818 . . . 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 2891 . . 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 440 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   E.wrex 2750   <.cop 3986   class class class wbr 4416   ` cfv 5601   NNcn 10637   EEcee 24967    Btwn cbtwn 24968  Cgrccgr 24969    InnerFiveSeg cifs 30851  Cgr3ccgr3 30852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-sup 7982  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-sum 13802  df-ee 24970  df-btwn 24971  df-cgr 24972  df-ofs 30799  df-ifs 30856  df-cgr3 30857
This theorem is referenced by:  colinearxfr  30891  brofs2  30893  brifs2  30894  endofsegid  30901  brsegle2  30925
  Copyright terms: Public domain W3C validator