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Theorem colineardim1 29285
Description: If  A is colinear with  B and  C, then 
A is in the same space as  B. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
colineardim1  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( A  Colinear  <. B ,  C >.  ->  A  e.  ( EE `  N ) ) )

Proof of Theorem colineardim1
Dummy variables  a 
b  c  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-colinear 29263 . . 3  |-  Colinear  =  `' { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) }
21breqi 4453 . 2  |-  ( A 
Colinear 
<. B ,  C >.  <->  A `' { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) } <. B ,  C >. )
3 simpr1 1002 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  A  e.  V )
4 opex 4711 . . . 4  |-  <. B ,  C >.  e.  _V
5 brcnvg 5181 . . . 4  |-  ( ( A  e.  V  /\  <. B ,  C >.  e. 
_V )  ->  ( A `' { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) } <. B ,  C >.  <->  <. B ,  C >. { <. <. b ,  c >. ,  a
>.  |  E. 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
) )
63, 4, 5sylancl 662 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( A `' { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) } <. B ,  C >.  <->  <. B ,  C >. { <. <. b ,  c >. ,  a
>.  |  E. 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
) )
7 df-br 4448 . . . 4  |-  ( <. B ,  C >. {
<. <. b ,  c
>. ,  a >.  |  E. 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 ,  C >. ,  A >.  e.  { <. <.
b ,  c >. ,  a >.  |  E. 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
>. ) ) } )
8 eleq1 2539 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
983anbi2d 1304 . . . . . . . . . 10  |-  ( b  =  B  ->  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  c  e.  ( EE `  n ) )  <->  ( a  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  c  e.  ( EE `  n ) ) ) )
10 opeq1 4213 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1110breq2d 4459 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
a  Btwn  <. b ,  c >.  <->  a  Btwn  <. B , 
c >. ) )
12 breq1 4450 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  Btwn  <. c ,  a >.  <->  B  Btwn  <. c ,  a >. )
)
13 opeq2 4214 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. a ,  b >.  =  <. a ,  B >. )
1413breq2d 4459 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
c  Btwn  <. a ,  b >.  <->  c  Btwn  <. a ,  B >. ) )
1511, 12, 143orbi123d 1298 . . . . . . . . . 10  |-  ( b  =  B  ->  (
( a  Btwn  <. b ,  c >.  \/  b  Btwn  <. c ,  a
>.  \/  c  Btwn  <. a ,  b >. )  <->  ( a  Btwn  <. B , 
c >.  \/  B  Btwn  <.
c ,  a >.  \/  c  Btwn  <. a ,  B >. ) ) )
169, 15anbi12d 710 . . . . . . . . 9  |-  ( b  =  B  ->  (
( ( 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  e.  ( EE
`  n )  /\  B  e.  ( EE `  n )  /\  c  e.  ( EE `  n
) )  /\  (
a  Btwn  <. B , 
c >.  \/  B  Btwn  <.
c ,  a >.  \/  c  Btwn  <. a ,  B >. ) ) ) )
1716rexbidv 2973 . . . . . . . 8  |-  ( b  =  B  ->  ( E. 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
>. ) )  <->  E. 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 >. ) ) ) )
18 eleq1 2539 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  e.  ( EE
`  n )  <->  C  e.  ( EE `  n ) ) )
19183anbi3d 1305 . . . . . . . . . 10  |-  ( c  =  C  ->  (
( a  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  c  e.  ( EE `  n ) )  <->  ( a  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) ) ) )
20 opeq2 4214 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2120breq2d 4459 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
a  Btwn  <. B , 
c >. 
<->  a  Btwn  <. B ,  C >. ) )
22 opeq1 4213 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. c ,  a >.  =  <. C ,  a >. )
2322breq2d 4459 . . . . . . . . . . 11  |-  ( c  =  C  ->  ( B  Btwn  <. c ,  a
>. 
<->  B  Btwn  <. C , 
a >. ) )
24 breq1 4450 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  Btwn  <. a ,  B >.  <->  C  Btwn  <. a ,  B >. ) )
2521, 23, 243orbi123d 1298 . . . . . . . . . 10  |-  ( c  =  C  ->  (
( a  Btwn  <. B , 
c >.  \/  B  Btwn  <.
c ,  a >.  \/  c  Btwn  <. a ,  B >. )  <->  ( a  Btwn  <. B ,  C >.  \/  B  Btwn  <. C , 
a >.  \/  C  Btwn  <.
a ,  B >. ) ) )
2619, 25anbi12d 710 . . . . . . . . 9  |-  ( c  =  C  ->  (
( ( 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  e.  ( EE
`  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n
) )  /\  (
a  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  a >.  \/  C  Btwn  <. a ,  B >. ) ) ) )
2726rexbidv 2973 . . . . . . . 8  |-  ( c  =  C  ->  ( E. 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 >. ) )  <->  E. 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 >. ) ) ) )
28 eleq1 2539 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
29283anbi1d 1303 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) )  <->  ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) ) ) )
30 breq1 4450 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  Btwn  <. B ,  C >. 
<->  A  Btwn  <. B ,  C >. ) )
31 opeq2 4214 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. C , 
a >.  =  <. C ,  A >. )
3231breq2d 4459 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( B  Btwn  <. C ,  a
>. 
<->  B  Btwn  <. C ,  A >. ) )
33 opeq1 4213 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  B >.  =  <. A ,  B >. )
3433breq2d 4459 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( C  Btwn  <. a ,  B >.  <-> 
C  Btwn  <. A ,  B >. ) )
3530, 32, 343orbi123d 1298 . . . . . . . . . 10  |-  ( a  =  A  ->  (
( a  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  a >.  \/  C  Btwn  <. a ,  B >. )  <->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
3629, 35anbi12d 710 . . . . . . . . 9  |-  ( a  =  A  ->  (
( ( 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  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  C  e.  ( EE `  n ) )  /\  ( A 
Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
3736rexbidv 2973 . . . . . . . 8  |-  ( a  =  A  ->  ( E. 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 >. ) )  <->  E. 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 >. ) ) ) )
3817, 27, 37eloprabg 6372 . . . . . . 7  |-  ( ( B  e.  ( EE
`  N )  /\  C  e.  W  /\  A  e.  V )  ->  ( <. <. B ,  C >. ,  A >.  e.  { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) }  <->  E. 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 >. ) ) ) )
39383comr 1204 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )  ->  ( <. <. B ,  C >. ,  A >.  e.  { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) }  <->  E. 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 >. ) ) ) )
4039adantl 466 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( <. <. B ,  C >. ,  A >.  e.  { <. <.
b ,  c >. ,  a >.  |  E. 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
>. ) ) }  <->  E. 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 >. ) ) ) )
41 simpl 457 . . . . . . 7  |-  ( ( ( 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  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n ) ) )
42 simp2 997 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )  ->  B  e.  ( EE `  N
) )
4342anim2i 569 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( N  e.  NN  /\  B  e.  ( EE `  N
) ) )
44 3simpa 993 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n
) )  ->  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) )
4544anim2i 569 . . . . . . . . 9  |-  ( ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n
) ) )  -> 
( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n ) ) ) )
46 axdimuniq 23889 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  B  e.  ( EE `  n
) ) )  ->  N  =  n )
4746adantrrl 723 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  N  =  n )
48 simprrl 763 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  A  e.  ( EE `  n ) )
49 fveq2 5864 . . . . . . . . . . . 12  |-  ( N  =  n  ->  ( EE `  N )  =  ( EE `  n
) )
5049eleq2d 2537 . . . . . . . . . . 11  |-  ( N  =  n  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( EE `  n ) ) )
5148, 50syl5ibrcom 222 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  ( N  =  n  ->  A  e.  ( EE `  N ) ) )
5247, 51mpd 15 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  A  e.  ( EE `  N ) )
5343, 45, 52syl2an 477 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  C  e.  ( EE `  n ) ) ) )  ->  A  e.  ( EE `  N ) )
5453exp32 605 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( n  e.  NN  ->  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  C  e.  ( EE `  n ) )  ->  A  e.  ( EE `  N ) ) ) )
5541, 54syl7 68 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( 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  e.  ( EE `  N ) ) ) )
5655rexlimdv 2953 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( E. 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  e.  ( EE `  N ) ) )
5740, 56sylbid 215 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( <. <. B ,  C >. ,  A >.  e.  { <. <.
b ,  c >. ,  a >.  |  E. 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  e.  ( EE `  N ) ) )
587, 57syl5bi 217 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( <. B ,  C >. { <. <.
b ,  c >. ,  a >.  |  E. 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  ->  A  e.  ( EE
`  N ) ) )
596, 58sylbid 215 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( A `' { <. <. b ,  c
>. ,  a >.  |  E. 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
>. ) ) } <. B ,  C >.  ->  A  e.  ( EE `  N
) ) )
602, 59syl5bi 217 1  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( A  Colinear  <. B ,  C >.  ->  A  e.  ( EE `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   <.cop 4033   class class class wbr 4447   `'ccnv 4998   ` cfv 5586   {coprab 6283   NNcn 10532   EEcee 23864    Btwn cbtwn 23865    Colinear ccolin 29261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-z 10861  df-uz 11079  df-fz 11669  df-ee 23867  df-colinear 29263
This theorem is referenced by:  liness  29369
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