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Theorem lineext 29954
Description: Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Assertion
Ref Expression
lineext  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( A  Colinear  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) )
Distinct variable groups:    f, N    A, f    B, f    C, f    D, f    f, E

Proof of Theorem lineext
StepHypRef Expression
1 brcolinear 29937 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
213adant3 1014 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
32anbi1d 702 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( A  Colinear  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. )  <->  ( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  /\  <. A ,  B >.Cgr
<. D ,  E >. ) ) )
4 simp1 994 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  N  e.  NN )
5 simp3r 1023 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  E  e.  ( EE `  N ) )
6 simp3l 1022 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  D  e.  ( EE `  N ) )
75, 6jca 530 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
8 simp21 1027 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
9 simp23 1029 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
108, 9jca 530 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
114, 7, 103jca 1174 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) ) )
1211adantr 463 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) ) )
13 axsegcon 24432 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. f  e.  ( EE `  N ) ( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. ) )
1412, 13syl 16 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  E. f  e.  ( EE `  N
) ( D  Btwn  <. E ,  f >.  /\ 
<. D ,  f >.Cgr <. A ,  C >. ) )
15 simprlr 762 . . . . . . . . . . 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 ) )  /\  ( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )  ->  <. A ,  B >.Cgr <. D ,  E >. )
16 simprrr 764 . . . . . . . . . . 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 ) )  /\  ( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )  ->  <. A ,  C >.Cgr <. D ,  f
>. )
17 an4 822 . . . . . . . . . . . . 13  |-  ( ( ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. )  /\  ( D 
Btwn  <. E ,  f
>.  /\  <. A ,  C >.Cgr
<. D ,  f >.
) )  <->  ( ( A  Btwn  <. B ,  C >.  /\  D  Btwn  <. E , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )
18 simpl1 997 . . . . . . . . . . . . . . . . 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 ) )  ->  N  e.  NN )
19 simpl21 1072 . . . . . . . . . . . . . . . . 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 ) )  ->  A  e.  ( EE `  N ) )
20 simpl22 1073 . . . . . . . . . . . . . . . . 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 ) )  ->  B  e.  ( EE `  N ) )
21 simpl3l 1049 . . . . . . . . . . . . . . . . 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 ) )  ->  D  e.  ( EE `  N ) )
22 simpl3r 1050 . . . . . . . . . . . . . . . . 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 ) )
23 cgrcomlr 29876 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  <->  <. B ,  A >.Cgr <. E ,  D >. ) )
2418, 19, 20, 21, 22, 23syl122anc 1235 . . . . . . . . . . . . . . . 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 ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  <->  <. B ,  A >.Cgr <. E ,  D >. ) )
2524anbi1d 702 . . . . . . . . . . . . . . 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 ) )  -> 
( ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  <->  ( <. B ,  A >.Cgr <. E ,  D >.  /\  <. A ,  C >.Cgr
<. D ,  f >.
) ) )
2625anbi2d 701 . . . . . . . . . . . . . 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 ) )  -> 
( ( ( A 
Btwn  <. B ,  C >.  /\  D  Btwn  <. E , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) )  <->  ( ( A 
Btwn  <. B ,  C >.  /\  D  Btwn  <. E , 
f >. )  /\  ( <. B ,  A >.Cgr <. E ,  D >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) ) )
27 simpl23 1074 . . . . . . . . . . . . . . 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 ) )  ->  C  e.  ( EE `  N ) )
28 simpr 459 . . . . . . . . . . . . . . 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 ) )  -> 
f  e.  ( EE
`  N ) )
29 cgrextend 29886 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N
)  /\  f  e.  ( EE `  N ) ) )  ->  (
( ( A  Btwn  <. B ,  C >.  /\  D  Btwn  <. E , 
f >. )  /\  ( <. B ,  A >.Cgr <. E ,  D >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) )  ->  <. B ,  C >.Cgr <. E ,  f
>. ) )
3018, 20, 19, 27, 22, 21, 28, 29syl133anc 1249 . . . . . . . . . . . . . 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 ) )  -> 
( ( ( A 
Btwn  <. B ,  C >.  /\  D  Btwn  <. E , 
f >. )  /\  ( <. B ,  A >.Cgr <. E ,  D >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) )  ->  <. B ,  C >.Cgr <. E ,  f
>. ) )
3126, 30sylbid 215 . . . . . . . . . . . . 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 ) )  -> 
( ( ( A 
Btwn  <. B ,  C >.  /\  D  Btwn  <. E , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) )  ->  <. B ,  C >.Cgr <. E ,  f
>. ) )
3217, 31syl5bi 217 . . . . . . . . . . . 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 ) )  -> 
( ( ( A 
Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr
<. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) )  ->  <. B ,  C >.Cgr <. E ,  f
>. ) )
3332imp 427 . . . . . . . . . . 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 ) )  /\  ( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )  ->  <. B ,  C >.Cgr <. E ,  f
>. )
3415, 16, 333jca 1174 . . . . . . . . . 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 ) )  /\  ( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) )
3534expr 613 . . . . . . . . 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 ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  (
( D  Btwn  <. E , 
f >.  /\  <. A ,  C >.Cgr <. D ,  f
>. )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) ) )
36 cgrcom 29868 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. D , 
f >.Cgr <. A ,  C >.  <->  <. A ,  C >.Cgr <. D ,  f >. ) )
3718, 21, 28, 19, 27, 36syl122anc 1235 . . . . . . . . . . 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 ) )  -> 
( <. D ,  f
>.Cgr <. A ,  C >.  <->  <. A ,  C >.Cgr <. D ,  f >. ) )
3837anbi2d 701 . . . . . . . . . 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 ) )  -> 
( ( D  Btwn  <. E ,  f >.  /\ 
<. D ,  f >.Cgr <. A ,  C >. )  <-> 
( D  Btwn  <. E , 
f >.  /\  <. A ,  C >.Cgr <. D ,  f
>. ) ) )
3938adantr 463 . . . . . . . . 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 ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  (
( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. )  <->  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )
40 simpl2 998 . . . . . . . . . . 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 ) )  -> 
( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
41 brcgr3 29924 . . . . . . . . . . 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 ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  f
>.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
4218, 40, 21, 22, 28, 41syl113anc 1238 . . . . . . . . . 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 ) )  -> 
( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.  <-> 
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) ) )
4342adantr 463 . . . . . . . . 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 ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  f
>.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
4435, 39, 433imtr4d 268 . . . . . . . 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 ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  (
( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
4544an32s 802 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  /\  f  e.  ( EE `  N
) )  ->  (
( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
4645reximdva 2929 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  ( E. f  e.  ( EE `  N ) ( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) )
4714, 46mpd 15 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
4847exp32 603 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  C >.  ->  ( <. A ,  B >.Cgr <. D ,  E >.  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) ) )
49 3ancoma 978 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  <->  ( B  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
50 btwncom 29892 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
5149, 50sylan2b 473 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
52513adant3 1014 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
53 simp3 996 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )
54 simp22 1028 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
55 axsegcon 24432 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. f  e.  ( EE `  N ) ( E  Btwn  <. D , 
f >.  /\  <. E , 
f >.Cgr <. B ,  C >. ) )
564, 53, 54, 9, 55syl112anc 1230 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  E. f  e.  ( EE `  N ) ( E  Btwn  <. D , 
f >.  /\  <. E , 
f >.Cgr <. B ,  C >. ) )
5756adantr 463 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  E. f  e.  ( EE `  N
) ( E  Btwn  <. D ,  f >.  /\ 
<. E ,  f >.Cgr <. B ,  C >. ) )
58 cgrextend 29886 . . . . . . . . . . . . 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 ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) )  ->  <. A ,  C >.Cgr <. D ,  f
>. ) )
5918, 40, 21, 22, 28, 58syl113anc 1238 . . . . . . . . . . . 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 ) )  -> 
( ( ( B 
Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) )  ->  <. A ,  C >.Cgr <. D ,  f
>. ) )
60 simpll 751 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  B >.Cgr <. D ,  E >. )
61 simpr 459 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  C >.Cgr <. D ,  f
>. )
62 simplr 753 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. B ,  C >.Cgr <. E ,  f
>. )
6360, 61, 623jca 1174 . . . . . . . . . . . . . 14  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  f
>.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) )
6463ex 432 . . . . . . . . . . . . 13  |-  ( (
<. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  ->  ( <. A ,  C >.Cgr <. D ,  f
>.  ->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr
<. D ,  f >.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
6564adantl 464 . . . . . . . . . . . 12  |-  ( ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) )  ->  ( <. A ,  C >.Cgr <. D ,  f >.  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) ) )
6659, 65sylcom 29 . . . . . . . . . . 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 ) )  -> 
( ( ( B 
Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  f
>.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
67 an4 822 . . . . . . . . . . . 12  |-  ( ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. )  /\  ( E 
Btwn  <. D ,  f
>.  /\  <. E ,  f
>.Cgr <. B ,  C >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. E ,  f >.Cgr <. B ,  C >. ) ) )
68 cgrcom 29868 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  f  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. E , 
f >.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <. E ,  f >. ) )
6918, 22, 28, 20, 27, 68syl122anc 1235 . . . . . . . . . . . . . 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 ,  f
>.Cgr <. B ,  C >.  <->  <. B ,  C >.Cgr <. E ,  f >. ) )
7069anbi2d 701 . . . . . . . . . . . . 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 ) )  -> 
( ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. E ,  f
>.Cgr <. B ,  C >. )  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
7170anbi2d 701 . . . . . . . . . . . 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 ) )  -> 
( ( ( B 
Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. E ,  f >.Cgr <. B ,  C >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) ) ) )
7267, 71syl5bb 257 . . . . . . . . . . 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 ) )  -> 
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr
<. D ,  E >. )  /\  ( E  Btwn  <. D ,  f >.  /\ 
<. E ,  f >.Cgr <. B ,  C >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  E  Btwn  <. D , 
f >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. ) ) ) )
7366, 72, 423imtr4d 268 . . . . . . . . . 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 ) )  -> 
( ( ( B 
Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr
<. D ,  E >. )  /\  ( E  Btwn  <. D ,  f >.  /\ 
<. E ,  f >.Cgr <. B ,  C >. ) )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
7473expdimp 435 . . . . . . . . 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 ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  (
( E  Btwn  <. D , 
f >.  /\  <. E , 
f >.Cgr <. B ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
7574an32s 802 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  /\  f  e.  ( EE `  N
) )  ->  (
( E  Btwn  <. D , 
f >.  /\  <. E , 
f >.Cgr <. B ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
7675reximdva 2929 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  ( E. f  e.  ( EE `  N ) ( E  Btwn  <. D , 
f >.  /\  <. E , 
f >.Cgr <. B ,  C >. )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) )
7757, 76mpd 15 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
7877exp32 603 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >.  ->  ( <. A ,  B >.Cgr <. D ,  E >.  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) ) )
7952, 78sylbird 235 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. C ,  A >.  ->  ( <. A ,  B >.Cgr <. D ,  E >.  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) ) )
80 cgrxfr 29933 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N ) ( f  Btwn  <. D ,  E >.  /\  <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. ) ) )
814, 8, 9, 54, 53, 80syl131anc 1239 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N ) ( f  Btwn  <. D ,  E >.  /\  <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. ) ) )
82 cgr3permute1 29926 . . . . . . . . . 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 ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.  <->  <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. ) )
8318, 40, 21, 22, 28, 82syl113anc 1238 . . . . . . . . 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 ) )  -> 
( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.  <->  <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. )
)
8483biimprd 223 . . . . . . . 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 ) )  -> 
( <. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >.  -> 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
8584adantld 465 . . . . . . 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 ) )  -> 
( ( f  Btwn  <. D ,  E >.  /\ 
<. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. )  -> 
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
8685reximdva 2929 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( E. f  e.  ( EE `  N
) ( f  Btwn  <. D ,  E >.  /\ 
<. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. )  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
8781, 86syld 44 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
8887expd 434 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  ( <. A ,  B >.Cgr <. D ,  E >.  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) ) )
8948, 79, 883jaod 1290 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
) )
9089impd 429 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  /\  <. A ,  B >.Cgr
<. D ,  E >. )  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
913, 90sylbid 215 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( A  Colinear  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    \/ w3o 970    /\ w3a 971    e. wcel 1823   E.wrex 2805   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24393    Btwn cbtwn 24394  Cgrccgr 24395  Cgr3ccgr3 29914    Colinear ccolin 29915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-ee 24396  df-btwn 24397  df-cgr 24398  df-ofs 29861  df-colinear 29917  df-cgr3 29919
This theorem is referenced by:  brsegle2  29987
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