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Theorem lineext 26003
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 25986 . . . 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 977 . . 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 686 . 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 957 . . . . . . . . 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 986 . . . . . . . . . 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 985 . . . . . . . . . 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 519 . . . . . . . . 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 990 . . . . . . . . . 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 992 . . . . . . . . . 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 519 . . . . . . . . 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 1134 . . . . . . . 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 452 . . . . . . 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 25859 . . . . . . 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 740 . . . . . . . . . . 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 742 . . . . . . . . . . 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 798 . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . . . . 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 25925 . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . 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 686 . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . 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 25935 . . . . . . . . . . . . . . 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 1207 . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . 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 209 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 1134 . . . . . . . . . 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 599 . . . . . . . . 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 25917 . . . . . . . . . . . 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 1193 . . . . . . . . . . 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 685 . . . . . . . . . 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 452 . . . . . . . . 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 961 . . . . . . . . . . 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 25973 . . . . . . . . . . 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 1196 . . . . . . . . . 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 452 . . . . . . . . 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 260 . . . . . . . 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 780 . . . . . . 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 2811 . . . . . 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 589 . . . 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 943 . . . . . . 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 25941 . . . . . . 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 462 . . . . . 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 977 . . . . 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 959 . . . . . . . . 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 991 . . . . . . . . 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 25859 . . . . . . . . 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 1188 . . . . . . . 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 452 . . . . . . 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 25935 . . . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  B >.Cgr <. D ,  E >. )
61 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  C >.Cgr <. D ,  f
>. )
62 simplr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. B ,  C >.Cgr <. E ,  f
>. )
6360, 61, 623jca 1134 . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 27 . . . . . . . . . . 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 798 . . . . . . . . . . . 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 25917 . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . 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 685 . . . . . . . . . . . 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 249 . . . . . . . . . . 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 260 . . . . . . . . . 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 427 . . . . . . . . 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 780 . . . . . . . 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 2811 . . . . . . 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 589 . . . . 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 227 . . . 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 25982 . . . . . . 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 1197 . . . . . 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 25975 . . . . . . . . . 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 1196 . . . . . . . . 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 215 . . . . . . . 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 454 . . . . . . 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 2811 . . . . . 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 42 . . . . 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 >. >. )
)
8887exp3a 426 . . . 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 1248 . . 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 >. >. )
) )
9089imp3a 421 . 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 207 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    e. wcel 1725   E.wrex 2699   <.cop 3810   class class class wbr 4205   ` cfv 5447   NNcn 9993   EEcee 25820    Btwn cbtwn 25821  Cgrccgr 25822  Cgr3ccgr3 25963    Colinear ccolin 25964
This theorem is referenced by:  brsegle2  26036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-oi 7472  df-card 7819  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-rp 10606  df-ico 10915  df-icc 10916  df-fz 11037  df-fzo 11129  df-seq 11317  df-exp 11376  df-hash 11612  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-clim 12275  df-sum 12473  df-ee 25823  df-btwn 25824  df-cgr 25825  df-ofs 25910  df-cgr3 25967  df-colinear 25968
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