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Theorem colinbtwnle 25960
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 25959 . . . . 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 25959 . . . . . . 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 25856 . . . . . . 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 25826 . . . . . . 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 25826 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  C >.Cgr <. C ,  A >. )
14 seglecgr12 25953 . . . . . . . 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 25901 . . . . 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 25857 . . . . . . . . . . 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 25856 . . . . . . . . . . . . . . 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 25959 . . . . . . . . . . . . . . . 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 25957 . . . . . . . . . . . . . 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 25928 . . . . . . . . . 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 25858 . . . . . . . . . . . . 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 4179 . . . . . . . . . . . 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 25959 . . . . . . . . . . . . . . 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 25957 . . . . . . . . . . . . . 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 25928 . . . . . . . . . 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 25854 . . . . . . . . . . . . 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 4179 . . . . . . . . . . . 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 1721   <.cop 3781   class class class wbr 4176   ` cfv 5417   NNcn 9960   EEcee 25735    Btwn cbtwn 25736  Cgrccgr 25737    Colinear ccolin 25879    Seg<_ csegle 25948
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-ee 25738  df-btwn 25739  df-cgr 25740  df-ofs 25825  df-ifs 25881  df-cgr3 25882  df-colinear 25883  df-segle 25949
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