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Theorem colinbtwnle 24741
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 24740 . . . . 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 945 . . . . . . 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 24740 . . . . . . 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 461 . . . . . 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 941 . . . . . . 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 24637 . . . . . . 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 461 . . . . . 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 443 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
9 simpr2 962 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
10 simpr3 963 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
118, 9, 10cgrrflx2d 24607 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. B ,  C >.Cgr <. C ,  B >. )
12 simpr1 961 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
138, 12, 10cgrrflx2d 24607 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  C >.Cgr <. C ,  A >. )
14 seglecgr12 24734 . . . . . . . 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 1214 . . . . . . 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 660 . . . . . 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 259 . . . . 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 519 . . . 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 451 . . 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 24682 . . . . 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 732 . . . . . . . . . . . 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 24638 . . . . . . . . . . 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 470 . . . . . . . . . . . . 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 696 . . . . . . . . . . . 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 24637 . . . . . . . . . . . . . . 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 939 . . . . . . . . . . . . . . . 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 24740 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . 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 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 ,  A >.  Seg<_  <. C ,  B >. )
32 segleantisym 24738 . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 660 . . . . . . . . . . 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 24709 . . . . . . . . . 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 24639 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  C >. )
38373adant3r2 1161 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  C >. )
39 breq1 4026 . . . . . . . . . . . 12  |-  ( B  =  A  ->  ( B  Btwn  <. A ,  C >.  <-> 
A  Btwn  <. A ,  C >. ) )
4038, 39syl5ibrcom 213 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  A  ->  B  Btwn  <. A ,  C >. ) )
4140adantr 451 . . . . . . . . . 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 14 . . . . . . . . 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 598 . . . . . . . 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 453 . . . . . . 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 423 . . . . . 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 214 . . . . . . 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 42 . . . . . 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 732 . . . . . . . . . . 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 733 . . . . . . . . . . . 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 943 . . . . . . . . . . . . . . 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 24740 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . 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 418 . . . . . . . . . . . . 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 697 . . . . . . . . . . . 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 24738 . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 660 . . . . . . . . . . 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 24709 . . . . . . . . . 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 24635 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  Btwn  <. A ,  C >. )
61603adant3r2 1161 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  Btwn  <. A ,  C >. )
62 breq1 4026 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( B  Btwn  <. A ,  C >.  <-> 
C  Btwn  <. A ,  C >. ) )
6361, 62syl5ibrcom 213 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  C  ->  B  Btwn  <. A ,  C >. ) )
6463adantr 451 . . . . . . . . . 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 14 . . . . . . . . 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 598 . . . . . . . 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 454 . . . . . . 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 423 . . . . . 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 1246 . . . . 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 206 . . . 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 418 . . 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 183 . 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 423 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 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517  Cgrccgr 24518    Colinear ccolin 24660    Seg<_ csegle 24729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 24519  df-btwn 24520  df-cgr 24521  df-ofs 24606  df-ifs 24662  df-cgr3 24663  df-colinear 24664  df-segle 24730
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