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Theorem colinbtwnle 28288
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 28287 . . . . 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 976 . . . . . . 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 28287 . . . . . . 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 475 . . . . . 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 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 ) ) )
6 btwncom 28184 . . . . . . 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 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 >. ) )
8 simpl 457 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
9 simpr2 995 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
10 simpr3 996 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
118, 9, 10cgrrflx2d 28154 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. B ,  C >.Cgr <. C ,  B >. )
12 simpr1 994 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
138, 12, 10cgrrflx2d 28154 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  C >.Cgr <. C ,  A >. )
14 seglecgr12 28281 . . . . . . . 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 1251 . . . . . . 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 679 . . . . . 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 268 . . . . 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 533 . . . 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 465 . . 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 28229 . . . . 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 755 . . . . . . . . . . . 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 28185 . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 715 . . . . . . . . . . . 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 28184 . . . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . . . 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 28287 . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . 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 716 . . . . . . . . . . . 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 28285 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 679 . . . . . . . . . . 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 28256 . . . . . . . . . 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 28186 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  C >. )
38373adant3r2 1198 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  C >. )
39 breq1 4398 . . . . . . . . . . . 12  |-  ( B  =  A  ->  ( B  Btwn  <. A ,  C >.  <-> 
A  Btwn  <. A ,  C >. ) )
4038, 39syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  A  ->  B  Btwn  <. A ,  C >. ) )
4140adantr 465 . . . . . . . . . 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 615 . . . . . . . 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 467 . . . . . . 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 434 . . . . . 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 223 . . . . . . 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 46 . . . . . 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 755 . . . . . . . . . . 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 756 . . . . . . . . . . . 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 974 . . . . . . . . . . . . . . 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 28287 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . 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 716 . . . . . . . . . . . 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 28285 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 679 . . . . . . . . . . 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 28256 . . . . . . . . . 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 28182 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  Btwn  <. A ,  C >. )
61603adant3r2 1198 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  Btwn  <. A ,  C >. )
62 breq1 4398 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( B  Btwn  <. A ,  C >.  <-> 
C  Btwn  <. A ,  C >. ) )
6361, 62syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  C  ->  B  Btwn  <. A ,  C >. ) )
6463adantr 465 . . . . . . . . . 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 615 . . . . . . . 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 468 . . . . . . 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 434 . . . . . 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 1283 . . . . 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 215 . . . 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 429 . . 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 191 . 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 434 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1370    e. wcel 1758   <.cop 3986   class class class wbr 4395   ` cfv 5521   NNcn 10428   EEcee 23281    Btwn cbtwn 23282  Cgrccgr 23283    Colinear ccolin 28207    Seg<_ csegle 28276
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-sum 13277  df-ee 23284  df-btwn 23285  df-cgr 23286  df-ofs 28153  df-colinear 28209  df-ifs 28210  df-cgr3 28211  df-segle 28277
This theorem is referenced by: (None)
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