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Theorem colineardim1 29873
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 29851 . . 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 4462 . 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 4720 . . . 4  |-  <. B ,  C >.  e.  _V
5 brcnvg 5193 . . . 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 4457 . . . 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 2529 . . . . . . . . . . 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 4219 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1110breq2d 4468 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
a  Btwn  <. b ,  c >.  <->  a  Btwn  <. B , 
c >. ) )
12 breq1 4459 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  Btwn  <. c ,  a >.  <->  B  Btwn  <. c ,  a >. )
)
13 opeq2 4220 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. a ,  b >.  =  <. a ,  B >. )
1413breq2d 4468 . . . . . . . . . . 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 2968 . . . . . . . 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 2529 . . . . . . . . . . 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 4220 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2120breq2d 4468 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
a  Btwn  <. B , 
c >. 
<->  a  Btwn  <. B ,  C >. ) )
22 opeq1 4219 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. c ,  a >.  =  <. C ,  a >. )
2322breq2d 4468 . . . . . . . . . . 11  |-  ( c  =  C  ->  ( B  Btwn  <. c ,  a
>. 
<->  B  Btwn  <. C , 
a >. ) )
24 breq1 4459 . . . . . . . . . . 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 2968 . . . . . . . 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 2529 . . . . . . . . . . 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 4459 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  Btwn  <. B ,  C >. 
<->  A  Btwn  <. B ,  C >. ) )
31 opeq2 4220 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. C , 
a >.  =  <. C ,  A >. )
3231breq2d 4468 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( B  Btwn  <. C ,  a
>. 
<->  B  Btwn  <. C ,  A >. ) )
33 opeq1 4219 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  B >.  =  <. A ,  B >. )
3433breq2d 4468 . . . . . . . . . . 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 2968 . . . . . . . 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 6389 . . . . . . 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 24342 . . . . . . . . . . 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 765 . . . . . . . . . . 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 5872 . . . . . . . . . . . 12  |-  ( N  =  n  ->  ( EE `  N )  =  ( EE `  n
) )
5049eleq2d 2527 . . . . . . . . . . 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 2947 . . . . 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 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109   <.cop 4038   class class class wbr 4456   `'ccnv 5007   ` cfv 5594   {coprab 6297   NNcn 10556   EEcee 24317    Btwn cbtwn 24318    Colinear ccolin 29849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-z 10886  df-uz 11107  df-fz 11698  df-ee 24320  df-colinear 29851
This theorem is referenced by:  liness  29957
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