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Theorem colineardim1 30840
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 30818 . . 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 4411 . 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 1015 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  A  e.  V )
4 opex 4667 . . . 4  |-  <. B ,  C >.  e.  _V
5 brcnvg 5018 . . . 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 669 . . 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 4406 . . . 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 2519 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
983anbi2d 1346 . . . . . . . . . 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 4169 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1110breq2d 4417 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
a  Btwn  <. b ,  c >.  <->  a  Btwn  <. B , 
c >. ) )
12 breq1 4408 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  Btwn  <. c ,  a >.  <->  B  Btwn  <. c ,  a >. )
)
13 opeq2 4170 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. a ,  b >.  =  <. a ,  B >. )
1413breq2d 4417 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
c  Btwn  <. a ,  b >.  <->  c  Btwn  <. a ,  B >. ) )
1511, 12, 143orbi123d 1340 . . . . . . . . . 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 718 . . . . . . . . 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 2903 . . . . . . . 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 2519 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  e.  ( EE
`  n )  <->  C  e.  ( EE `  n ) ) )
19183anbi3d 1347 . . . . . . . . . 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 4170 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2120breq2d 4417 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
a  Btwn  <. B , 
c >. 
<->  a  Btwn  <. B ,  C >. ) )
22 opeq1 4169 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. c ,  a >.  =  <. C ,  a >. )
2322breq2d 4417 . . . . . . . . . . 11  |-  ( c  =  C  ->  ( B  Btwn  <. c ,  a
>. 
<->  B  Btwn  <. C , 
a >. ) )
24 breq1 4408 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  Btwn  <. a ,  B >.  <->  C  Btwn  <. a ,  B >. ) )
2521, 23, 243orbi123d 1340 . . . . . . . . . 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 718 . . . . . . . . 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 2903 . . . . . . . 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 2519 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
29283anbi1d 1345 . . . . . . . . . 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 4408 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  Btwn  <. B ,  C >. 
<->  A  Btwn  <. B ,  C >. ) )
31 opeq2 4170 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. C , 
a >.  =  <. C ,  A >. )
3231breq2d 4417 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( B  Btwn  <. C ,  a
>. 
<->  B  Btwn  <. C ,  A >. ) )
33 opeq1 4169 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  B >.  =  <. A ,  B >. )
3433breq2d 4417 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( C  Btwn  <. a ,  B >.  <-> 
C  Btwn  <. A ,  B >. ) )
3530, 32, 343orbi123d 1340 . . . . . . . . . 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 718 . . . . . . . . 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 2903 . . . . . . . 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 1217 . . . . . 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 468 . . . . 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 459 . . . . . . 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 1010 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )  ->  B  e.  ( EE `  N
) )
4342anim2i 573 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( N  e.  NN  /\  B  e.  ( EE `  N
) ) )
44 3simpa 1006 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n
) )  ->  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) )
4544anim2i 573 . . . . . . . . 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 24955 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  B  e.  ( EE `  n
) ) )  ->  N  =  n )
4746adantrrl 731 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  N  =  n )
48 simprrl 775 . . . . . . . . . . 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 5870 . . . . . . . . . . . 12  |-  ( N  =  n  ->  ( EE `  N )  =  ( EE `  n
) )
5049eleq2d 2516 . . . . . . . . . . 11  |-  ( N  =  n  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( EE `  n ) ) )
5148, 50syl5ibrcom 226 . . . . . . . . . 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 480 . . . . . . . 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 610 . . . . . . 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 70 . . . . . 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 2879 . . . . 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 219 . . . 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 221 . . 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 219 . 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 221 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 188    /\ wa 371    \/ w3o 985    /\ w3a 986    = wceq 1446    e. wcel 1889   E.wrex 2740   _Vcvv 3047   <.cop 3976   class class class wbr 4405   `'ccnv 4836   ` cfv 5585   {coprab 6296   NNcn 10616   EEcee 24930    Btwn cbtwn 24931    Colinear ccolin 30816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-z 10945  df-uz 11167  df-fz 11792  df-ee 24933  df-colinear 30818
This theorem is referenced by:  liness  30924
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