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Theorem lineext 28241
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 28224 . . . 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 1008 . . 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 704 . 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 988 . . . . . . . . 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 1017 . . . . . . . . . 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 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
) ) )  ->  D  e.  ( EE `  N ) )
75, 6jca 532 . . . . . . . . 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 1021 . . . . . . . . . 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 1023 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
108, 9jca 532 . . . . . . . . 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 1168 . . . . . . . 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 465 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( 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 23308 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( E  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  E. f  e.  ( EE `  N ) ( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. ) )
1412, 13syl 16 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  E. f  e.  ( EE `  N
) ( D  Btwn  <. E ,  f >.  /\ 
<. D ,  f >.Cgr <. A ,  C >. ) )
15 simprlr 762 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  f  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )  ->  <. A ,  B >.Cgr <. D ,  E >. )
16 simprrr 764 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  f  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  /\  ( D  Btwn  <. E ,  f >.  /\ 
<. A ,  C >.Cgr <. D ,  f >. ) ) )  ->  <. A ,  C >.Cgr <. D ,  f
>. )
17 an4 820 . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . . . . 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 1066 . . . . . . . . . . . . . . . . 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 1067 . . . . . . . . . . . . . . . . 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 1043 . . . . . . . . . . . . . . . . 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 1044 . . . . . . . . . . . . . . . . 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 28163 . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . 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 704 . . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 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 28173 . . . . . . . . . . . . . . 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 1242 . . . . . . . . . . . . . 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 1168 . . . . . . . . . 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 615 . . . . . . . . 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 28155 . . . . . . . . . . . 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 1228 . . . . . . . . . . 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 703 . . . . . . . . . 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 465 . . . . . . . . 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 992 . . . . . . . . . . 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 28211 . . . . . . . . . . 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 1231 . . . . . . . . . 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 465 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  f  e.  ( EE `  N ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  f
>.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
4435, 39, 433imtr4d 268 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  f  e.  ( EE `  N ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  ->  (
( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
4544an32s 802 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  /\  f  e.  ( EE `  N
) )  ->  (
( D  Btwn  <. E , 
f >.  /\  <. D , 
f >.Cgr <. A ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
4645reximdva 2924 . . . . . 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 605 . . . 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 972 . . . . . . 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 28179 . . . . . . 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 475 . . . . . 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 1008 . . . . 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 990 . . . . . . . . 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 1022 . . . . . . . . 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 23308 . . . . . . . . 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 1223 . . . . . . . 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 465 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( 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 28173 . . . . . . . . . . . . 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 1231 . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  B >.Cgr <. D ,  E >. )
61 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. A ,  C >.Cgr <. D ,  f
>. )
62 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  /\  <. A ,  C >.Cgr
<. D ,  f >.
)  ->  <. B ,  C >.Cgr <. E ,  f
>. )
6360, 61, 623jca 1168 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 820 . . . . . . . . . . . 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 28155 . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . 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 703 . . . . . . . . . . . . 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 703 . . . . . . . . . . . 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 802 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. D ,  E >. ) )  /\  f  e.  ( EE `  N
) )  ->  (
( E  Btwn  <. D , 
f >.  /\  <. E , 
f >.Cgr <. B ,  C >. )  ->  <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E , 
f >. >. ) )
7675reximdva 2924 . . . . . . 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 605 . . . . 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 28220 . . . . . . 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 1232 . . . . . 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 28213 . . . . . . . . . 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 1231 . . . . . . . . 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 467 . . . . . . 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 2924 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( E. f  e.  ( EE `  N
) ( f  Btwn  <. D ,  E >.  /\ 
<. A ,  <. C ,  B >. >.Cgr3 <. D ,  <. f ,  E >. >. )  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
8781, 86syld 44 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\ 
<. A ,  B >.Cgr <. D ,  E >. )  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
)
8887expd 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 1283 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  ->  E. f  e.  ( EE `  N )
<. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  f >. >. )
) )
9089impd 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 964    /\ w3a 965    e. wcel 1758   E.wrex 2796   <.cop 3981   class class class wbr 4390   ` cfv 5516   NNcn 10423   EEcee 23269    Btwn cbtwn 23270  Cgrccgr 23271  Cgr3ccgr3 28201    Colinear ccolin 28202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-ee 23272  df-btwn 23273  df-cgr 23274  df-ofs 28148  df-colinear 28204  df-cgr3 28206
This theorem is referenced by:  brsegle2  28274
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