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Theorem colineardim1 24756
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 24736 . . 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 4045 . 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 961 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  A  e.  V )
4 opex 4253 . . . 4  |-  <. B ,  C >.  e.  _V
5 brcnvg 4878 . . . 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 643 . . 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 4040 . . . 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 2356 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
983anbi2d 1257 . . . . . . . . . 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 3812 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1110breq2d 4051 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
a  Btwn  <. b ,  c >.  <->  a  Btwn  <. B , 
c >. ) )
12 breq1 4042 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  Btwn  <. c ,  a >.  <->  B  Btwn  <. c ,  a >. )
)
13 opeq2 3813 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. a ,  b >.  =  <. a ,  B >. )
1413breq2d 4051 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
c  Btwn  <. a ,  b >.  <->  c  Btwn  <. a ,  B >. ) )
1511, 12, 143orbi123d 1251 . . . . . . . . . 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 691 . . . . . . . . 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 2577 . . . . . . . 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 2356 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  e.  ( EE
`  n )  <->  C  e.  ( EE `  n ) ) )
19183anbi3d 1258 . . . . . . . . . 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 3813 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2120breq2d 4051 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
a  Btwn  <. B , 
c >. 
<->  a  Btwn  <. B ,  C >. ) )
22 opeq1 3812 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. c ,  a >.  =  <. C ,  a >. )
2322breq2d 4051 . . . . . . . . . . 11  |-  ( c  =  C  ->  ( B  Btwn  <. c ,  a
>. 
<->  B  Btwn  <. C , 
a >. ) )
24 breq1 4042 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  Btwn  <. a ,  B >.  <->  C  Btwn  <. a ,  B >. ) )
2521, 23, 243orbi123d 1251 . . . . . . . . . 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 691 . . . . . . . . 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 2577 . . . . . . . 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 2356 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
29283anbi1d 1256 . . . . . . . . . 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 4042 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  Btwn  <. B ,  C >. 
<->  A  Btwn  <. B ,  C >. ) )
31 opeq2 3813 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. C , 
a >.  =  <. C ,  A >. )
3231breq2d 4051 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( B  Btwn  <. C ,  a
>. 
<->  B  Btwn  <. C ,  A >. ) )
33 opeq1 3812 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  B >.  =  <. A ,  B >. )
3433breq2d 4051 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( C  Btwn  <. a ,  B >.  <-> 
C  Btwn  <. A ,  B >. ) )
3530, 32, 343orbi123d 1251 . . . . . . . . . 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 691 . . . . . . . . 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 2577 . . . . . . . 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 5951 . . . . . . 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 1159 . . . . . 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 452 . . . . 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 443 . . . . . . 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 956 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )  ->  B  e.  ( EE `  N
) )
4342anim2i 552 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( N  e.  NN  /\  B  e.  ( EE `  N
) ) )
44 3simpa 952 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n
) )  ->  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) )
4544anim2i 552 . . . . . . . . 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 24613 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  B  e.  ( EE `  n
) ) )  ->  N  =  n )
4746adantrrl 704 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  N  =  n )
48 simprrl 740 . . . . . . . . . . 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 5541 . . . . . . . . . . . 12  |-  ( N  =  n  ->  ( EE `  N )  =  ( EE `  n
) )
5049eleq2d 2363 . . . . . . . . . . 11  |-  ( N  =  n  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( EE `  n ) ) )
5148, 50syl5ibrcom 213 . . . . . . . . . 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 14 . . . . . . . . 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 463 . . . . . . . 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 588 . . . . . . 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 63 . . . . . 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 2679 . . . . 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 206 . . . 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 208 . . 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 206 . 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 208 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   <.cop 3656   class class class wbr 4039   `'ccnv 4704   ` cfv 5271   {coprab 5875   NNcn 9762   EEcee 24588    Btwn cbtwn 24589    Colinear ccolin 24732
This theorem is referenced by:  liness  24840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-z 10041  df-uz 10247  df-fz 10799  df-ee 24591  df-colinear 24736
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