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Theorem colinbtwnle 25759
Description: Given three colinear points  A,  B, and  C,  B falls in the middle iff the two segments to 
B are no longer than  A C. Theorem 5.12 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
colinbtwnle  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >.  ->  ( B  Btwn  <. A ,  C >.  <-> 
( <. A ,  B >. 
Seg<_ 
<. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. ) ) ) )

Proof of Theorem colinbtwnle
StepHypRef Expression
1 btwnsegle 25758 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >.  ->  <. A ,  B >.  Seg<_  <. A ,  C >. ) )
2 3anrev 947 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  <->  ( C  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )
3 btwnsegle 25758 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. C ,  A >.  ->  <. C ,  B >.  Seg<_  <. C ,  A >. ) )
42, 3sylan2b 462 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. C ,  A >.  ->  <. C ,  B >.  Seg<_  <. C ,  A >. ) )
5 3ancoma 943 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  <->  ( B  e.  ( EE `  N
)  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
6 btwncom 25655 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
75, 6sylan2b 462 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
8 simpl 444 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
9 simpr2 964 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
10 simpr3 965 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
118, 9, 10cgrrflx2d 25625 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. B ,  C >.Cgr <. C ,  B >. )
12 simpr1 963 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
138, 12, 10cgrrflx2d 25625 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  C >.Cgr <. C ,  A >. )
14 seglecgr12 25752 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  A  e.  ( EE `  N ) ) )  ->  (
( <. B ,  C >.Cgr
<. C ,  B >.  /\ 
<. A ,  C >.Cgr <. C ,  A >. )  ->  ( <. B ,  C >.  Seg<_  <. A ,  C >.  <->  <. C ,  B >.  Seg<_  <. C ,  A >. ) ) )
158, 9, 10, 12, 10, 10, 9, 10, 12, 14syl333anc 1216 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( <. B ,  C >.Cgr <. C ,  B >.  /\  <. A ,  C >.Cgr
<. C ,  A >. )  ->  ( <. B ,  C >.  Seg<_  <. A ,  C >.  <->  <. C ,  B >.  Seg<_  <. C ,  A >. ) ) )
1611, 13, 15mp2and 661 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. B ,  C >. 
Seg<_ 
<. A ,  C >.  <->  <. C ,  B >.  Seg<_  <. C ,  A >. ) )
174, 7, 163imtr4d 260 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >.  ->  <. B ,  C >.  Seg<_  <. A ,  C >. ) )
181, 17jcad 520 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >.  ->  ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\  <. B ,  C >.  Seg<_  <. A ,  C >. ) ) )
1918adantr 452 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A  Colinear  <. B ,  C >. )  ->  ( B  Btwn  <. A ,  C >.  -> 
( <. A ,  B >. 
Seg<_ 
<. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. ) ) )
20 brcolinear 25700 . . . . 5  |-  ( ( 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 >. ) ) )
21 simprl 733 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  A  Btwn  <. B ,  C >. )
228, 12, 9, 10, 21btwncomand 25656 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  A  Btwn  <. C ,  B >. )
2316biimpa 471 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  <. B ,  C >.  Seg<_  <. A ,  C >. )  ->  <. C ,  B >.  Seg<_  <. C ,  A >. )
2423adantrl 697 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  <. C ,  B >.  Seg<_  <. C ,  A >. )
25 btwncom 25655 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  C >. 
<->  A  Btwn  <. C ,  B >. ) )
26 3anrot 941 . . . . . . . . . . . . . . . 16  |-  ( ( C  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
27 btwnsegle 25758 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. C ,  B >.  ->  <. C ,  A >.  Seg<_  <. C ,  B >. ) )
2826, 27sylan2br 463 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. C ,  B >.  ->  <. C ,  A >.  Seg<_  <. C ,  B >. ) )
2925, 28sylbid 207 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  C >.  ->  <. C ,  A >.  Seg<_  <. C ,  B >. ) )
3029imp 419 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A  Btwn  <. B ,  C >. )  ->  <. C ,  A >.  Seg<_  <. C ,  B >. )
3130adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  <. C ,  A >.  Seg<_  <. C ,  B >. )
32 segleantisym 25756 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )  ->  ( ( <. C ,  B >.  Seg<_  <. C ,  A >.  /\ 
<. C ,  A >.  Seg<_  <. C ,  B >. )  ->  <. C ,  B >.Cgr
<. C ,  A >. ) )
338, 10, 9, 10, 12, 32syl122anc 1193 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( <. C ,  B >.  Seg<_  <. C ,  A >.  /\  <. C ,  A >. 
Seg<_ 
<. C ,  B >. )  ->  <. C ,  B >.Cgr
<. C ,  A >. ) )
3433adantr 452 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  ( ( <. C ,  B >.  Seg<_  <. C ,  A >.  /\ 
<. C ,  A >.  Seg<_  <. C ,  B >. )  ->  <. C ,  B >.Cgr
<. C ,  A >. ) )
3524, 31, 34mp2and 661 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  <. C ,  B >.Cgr <. C ,  A >. )
368, 10, 9, 12, 22, 35endofsegidand 25727 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  B  =  A )
37 btwntriv1 25657 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  C >. )
38373adant3r2 1163 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  C >. )
39 breq1 4149 . . . . . . . . . . . 12  |-  ( B  =  A  ->  ( B  Btwn  <. A ,  C >.  <-> 
A  Btwn  <. A ,  C >. ) )
4038, 39syl5ibrcom 214 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  A  ->  B  Btwn  <. A ,  C >. ) )
4140adantr 452 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  ( B  =  A  ->  B  Btwn  <. A ,  C >. ) )
4236, 41mpd 15 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( A  Btwn  <. B ,  C >.  /\  <. B ,  C >. 
Seg<_ 
<. A ,  C >. ) )  ->  B  Btwn  <. A ,  C >. )
4342expr 599 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A  Btwn  <. B ,  C >. )  ->  ( <. B ,  C >.  Seg<_  <. A ,  C >.  ->  B  Btwn  <. A ,  C >. ) )
4443adantld 454 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A  Btwn  <. B ,  C >. )  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) )
4544ex 424 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Btwn  <. B ,  C >.  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) ) )
467biimprd 215 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. C ,  A >.  ->  B  Btwn  <. A ,  C >. ) )
4746a1dd 44 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. C ,  A >.  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) ) )
48 simprl 733 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  C  Btwn  <. A ,  B >. )
49 simprr 734 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  <. A ,  B >.  Seg<_  <. A ,  C >. )
50 3ancomb 945 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  <->  ( A  e.  ( EE `  N
)  /\  C  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )
51 btwnsegle 25758 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  <. A ,  C >.  Seg<_  <. A ,  B >. ) )
5250, 51sylan2b 462 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  <. A ,  C >.  Seg<_  <. A ,  B >. ) )
5352imp 419 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  C  Btwn  <. A ,  B >. )  ->  <. A ,  C >.  Seg<_  <. A ,  B >. )
5453adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  <. A ,  C >.  Seg<_  <. A ,  B >. )
55 segleantisym 25756 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. A ,  C >.  Seg<_  <. A ,  B >. )  ->  <. A ,  B >.Cgr
<. A ,  C >. ) )
568, 12, 9, 12, 10, 55syl122anc 1193 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\  <. A ,  C >. 
Seg<_ 
<. A ,  B >. )  ->  <. A ,  B >.Cgr
<. A ,  C >. ) )
5756adantr 452 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. A ,  C >.  Seg<_  <. A ,  B >. )  ->  <. A ,  B >.Cgr
<. A ,  C >. ) )
5849, 54, 57mp2and 661 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  <. A ,  B >.Cgr <. A ,  C >. )
598, 12, 9, 10, 48, 58endofsegidand 25727 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  B  =  C )
60 btwntriv2 25653 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  Btwn  <. A ,  C >. )
61603adant3r2 1163 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  Btwn  <. A ,  C >. )
62 breq1 4149 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( B  Btwn  <. A ,  C >.  <-> 
C  Btwn  <. A ,  C >. ) )
6361, 62syl5ibrcom 214 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  C  ->  B  Btwn  <. A ,  C >. ) )
6463adantr 452 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  ( B  =  C  ->  B  Btwn  <. A ,  C >. ) )
6559, 64mpd 15 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  ( C  Btwn  <. A ,  B >.  /\  <. A ,  B >. 
Seg<_ 
<. A ,  C >. ) )  ->  B  Btwn  <. A ,  C >. )
6665expr 599 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  C  Btwn  <. A ,  B >. )  ->  ( <. A ,  B >.  Seg<_  <. A ,  C >.  ->  B  Btwn  <. A ,  C >. ) )
6766adantrd 455 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  C  Btwn  <. A ,  B >. )  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) )
6867ex 424 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) ) )
6945, 47, 683jaod 1248 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) ) )
7020, 69sylbid 207 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >.  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) ) )
7170imp 419 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A  Colinear  <. B ,  C >. )  ->  ( ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. ) )
7219, 71impbid 184 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  A  Colinear  <. B ,  C >. )  ->  ( B  Btwn  <. A ,  C >.  <->  ( <. A ,  B >.  Seg<_  <. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. ) ) )
7372ex 424 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >.  ->  ( B  Btwn  <. A ,  C >.  <-> 
( <. A ,  B >. 
Seg<_ 
<. A ,  C >.  /\ 
<. B ,  C >.  Seg<_  <. A ,  C >. ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146   ` cfv 5387   NNcn 9925   EEcee 25534    Btwn cbtwn 25535  Cgrccgr 25536    Colinear ccolin 25678    Seg<_ csegle 25747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-ee 25537  df-btwn 25538  df-cgr 25539  df-ofs 25624  df-ifs 25680  df-cgr3 25681  df-colinear 25682  df-segle 25748
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