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Theorem colinbtwnle 23915
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 23914 . . . . 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 950 . . . . . . 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 23914 . . . . . . 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 463 . . . . . 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 946 . . . . . . 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 23811 . . . . . . 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 463 . . . . . 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 445 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
9 simpr2 967 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
10 simpr3 968 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
118, 9, 10cgrrflx2d 23781 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. B ,  C >.Cgr <. C ,  B >. )
12 simpr1 966 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
138, 12, 10cgrrflx2d 23781 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  C >.Cgr <. C ,  A >. )
14 seglecgr12 23908 . . . . . . . 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 1219 . . . . . . 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 663 . . . . . 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 261 . . . . 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 521 . . . 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 453 . . 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 23856 . . . . 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 735 . . . . . . . . . . . 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 23812 . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 699 . . . . . . . . . . . 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 23811 . . . . . . . . . . . . . . 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 944 . . . . . . . . . . . . . . . 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 23914 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . 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 700 . . . . . . . . . . . 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 23912 . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 663 . . . . . . . . . . 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 23883 . . . . . . . . . 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 23813 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  A  Btwn  <. A ,  C >. )
38373adant3r2 1166 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  Btwn  <. A ,  C >. )
39 breq1 3923 . . . . . . . . . . . 12  |-  ( B  =  A  ->  ( B  Btwn  <. A ,  C >.  <-> 
A  Btwn  <. A ,  C >. ) )
4038, 39syl5ibrcom 215 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  A  ->  B  Btwn  <. A ,  C >. ) )
4140adantr 453 . . . . . . . . . 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 16 . . . . . . . . 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 601 . . . . . . . 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 455 . . . . . . 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 425 . . . . . 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 216 . . . . . . 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 735 . . . . . . . . . . 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 736 . . . . . . . . . . . 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 948 . . . . . . . . . . . . . . 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 23914 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . 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 700 . . . . . . . . . . . 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 23912 . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 663 . . . . . . . . . . 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 23883 . . . . . . . . . 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 23809 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  C  Btwn  <. A ,  C >. )
61603adant3r2 1166 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  Btwn  <. A ,  C >. )
62 breq1 3923 . . . . . . . . . . . 12  |-  ( B  =  C  ->  ( B  Btwn  <. A ,  C >.  <-> 
C  Btwn  <. A ,  C >. ) )
6361, 62syl5ibrcom 215 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  =  C  ->  B  Btwn  <. A ,  C >. ) )
6463adantr 453 . . . . . . . . . 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 16 . . . . . . . . 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 601 . . . . . . . 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 456 . . . . . . 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 425 . . . . . 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 1251 . . . . 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 208 . . . 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 420 . . 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 185 . 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 425 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 6    <-> wb 178    /\ wa 360    \/ w3o 938    /\ w3a 939    = wceq 1619    e. wcel 1621   <.cop 3547   class class class wbr 3920   ` cfv 4592   NNcn 9626   EEcee 23690    Btwn cbtwn 23691  Cgrccgr 23692    Colinear ccolin 23834    Seg<_ csegle 23903
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-ee 23693  df-btwn 23694  df-cgr 23695  df-ofs 23780  df-ifs 23836  df-cgr3 23837  df-colinear 23838  df-segle 23904
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