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Theorem lineext 23873
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 23856 . . . 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 980 . . 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 688 . 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 960 . . . . . . . . 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 989 . . . . . . . . . 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 988 . . . . . . . . . 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 520 . . . . . . . . 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 993 . . . . . . . . . 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 995 . . . . . . . . . 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 520 . . . . . . . . 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 1137 . . . . . . . 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 453 . . . . . . 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 23729 . . . . . . 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 17 . . . . . 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 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 ,  B >.Cgr <. D ,  E >. )
16 simprrr 744 . . . . . . . . . . 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 800 . . . . . . . . . . . . 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 963 . . . . . . . . . . . . . . . . 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 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 ) )  ->  A  e.  ( EE `  N ) )
20 simpl22 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 ) )  ->  B  e.  ( EE `  N ) )
21 simpl3l 1015 . . . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . . . 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 23795 . . . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . 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 1040 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . 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 23805 . . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 420 . . . . . . . . . . 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 1137 . . . . . . . . . 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 601 . . . . . . . . 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 23787 . . . . . . . . . . . 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 1196 . . . . . . . . . . 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 687 . . . . . . . . . 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 453 . . . . . . . . 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 964 . . . . . . . . . . 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 23843 . . . . . . . . . . 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 1199 . . . . . . . . . 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 453 . . . . . . . . 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 261 . . . . . . . 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 782 . . . . . . 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 2617 . . . . . 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 16 . . . . 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 591 . . . 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 946 . . . . . . 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 23811 . . . . . . 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 463 . . . . . 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 980 . . . . 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 962 . . . . . . . . 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 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
) ) )  ->  B  e.  ( EE `  N ) )
55 axsegcon 23729 . . . . . . . . 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 1191 . . . . . . . 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 453 . . . . . . 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 23805 . . . . . . . . . . . . 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 1199 . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  B >.Cgr <. D ,  E >. )
61 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  C >.Cgr <. D ,  f
>. )
62 simplr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. B ,  C >.Cgr <. E ,  f
>. )
6360, 61, 623jca 1137 . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . 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 454 . . . . . . . . . . . 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 800 . . . . . . . . . . . 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 23787 . . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . 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 687 . . . . . . . . . . . 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 250 . . . . . . . . . . 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 261 . . . . . . . . . 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 428 . . . . . . . . 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 782 . . . . . . . 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 2617 . . . . . . 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 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
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  E. f  e.  ( EE `  N
) <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
7877exp32 591 . . . . 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 228 . . . 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 23852 . . . . . . 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 1200 . . . . . 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 23845 . . . . . . . . . 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 1199 . . . . . . . . 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 216 . . . . . . . 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 455 . . . . . . 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 2617 . . . . . 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 427 . . . 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 1251 . . 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 422 . 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 208 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 6    <-> wb 178    /\ wa 360    \/ w3o 938    /\ w3a 939    e. wcel 1621   E.wrex 2510   <.cop 3547   class class class wbr 3920   ` cfv 4592   NNcn 9626   EEcee 23690    Btwn cbtwn 23691  Cgrccgr 23692  Cgr3ccgr3 23833    Colinear ccolin 23834
This theorem is referenced by:  brsegle2  23906
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-ee 23693  df-btwn 23694  df-cgr 23695  df-ofs 23780  df-cgr3 23837  df-colinear 23838
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