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Theorem lineext 28036
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 28019 . . . 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 1003 . . 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 699 . 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 983 . . . . . . . . 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 1012 . . . . . . . . . 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 1011 . . . . . . . . . 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 529 . . . . . . . . 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 1016 . . . . . . . . . 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 1018 . . . . . . . . . 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 529 . . . . . . . . 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 1163 . . . . . . . 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 462 . . . . . . 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 23108 . . . . . . 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 757 . . . . . . . . . . 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 759 . . . . . . . . . . 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 815 . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . 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 1061 . . . . . . . . . . . . . . . . 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 1062 . . . . . . . . . . . . . . . . 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 1038 . . . . . . . . . . . . . . . . 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 1039 . . . . . . . . . . . . . . . . 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 27958 . . . . . . . . . . . . . . . . 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 1222 . . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . 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 1063 . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  f  e.  ( EE `  N ) )  -> 
f  e.  ( EE
`  N ) )
29 cgrextend 27968 . . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . . 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 429 . . . . . . . . . . 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 1163 . . . . . . . . . 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 612 . . . . . . . . 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 27950 . . . . . . . . . . . 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 1222 . . . . . . . . . . 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 698 . . . . . . . . . 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 462 . . . . . . . . 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 987 . . . . . . . . . . 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 28006 . . . . . . . . . . 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 1225 . . . . . . . . . 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 462 . . . . . . . . 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 797 . . . . . . 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 2826 . . . . . 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 602 . . . 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 967 . . . . . . 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 27974 . . . . . . 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 472 . . . . . 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 1003 . . . . 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 985 . . . . . . . . 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 1017 . . . . . . . . 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 23108 . . . . . . . . 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 1217 . . . . . . . 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 462 . . . . . . 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 27968 . . . . . . . . . . . . 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 1225 . . . . . . . . . . . 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 748 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  B >.Cgr <. D ,  E >. )
61 simpr 458 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  C >.Cgr <. D ,  f
>. )
62 simplr 749 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. B ,  C >.Cgr <. E ,  f
>. )
6360, 61, 623jca 1163 . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 815 . . . . . . . . . . . 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 27950 . . . . . . . . . . . . . . 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 1222 . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . 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 698 . . . . . . . . . . . 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 437 . . . . . . . . 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 797 . . . . . . . 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 2826 . . . . . . 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 602 . . . . 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 28015 . . . . . . 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 1226 . . . . . 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 28008 . . . . . . . . . 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 1225 . . . . . . . . 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 464 . . . . . . 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 2826 . . . . . 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 >. >. )
)
8887exp3a 436 . . . 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 1277 . . 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 431 . 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 369    \/ w3o 959    /\ w3a 960    e. wcel 1761   E.wrex 2714   <.cop 3880   class class class wbr 4289   ` cfv 5415   NNcn 10318   EEcee 23069    Btwn cbtwn 23070  Cgrccgr 23071  Cgr3ccgr3 27996    Colinear ccolin 27997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-ee 23072  df-btwn 23073  df-cgr 23074  df-ofs 27943  df-colinear 27999  df-cgr3 28001
This theorem is referenced by:  brsegle2  28069
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