Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  colineardim1 Structured version   Unicode version

Theorem colineardim1 30610
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 30588 . . 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 4423 . 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 1011 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  A  e.  V )
4 opex 4677 . . . 4  |-  <. B ,  C >.  e.  _V
5 brcnvg 5026 . . . 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 666 . . 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 4418 . . . 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 2492 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
983anbi2d 1340 . . . . . . . . . 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 4181 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1110breq2d 4429 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
a  Btwn  <. b ,  c >.  <->  a  Btwn  <. B , 
c >. ) )
12 breq1 4420 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
b  Btwn  <. c ,  a >.  <->  B  Btwn  <. c ,  a >. )
)
13 opeq2 4182 . . . . . . . . . . . 12  |-  ( b  =  B  ->  <. a ,  b >.  =  <. a ,  B >. )
1413breq2d 4429 . . . . . . . . . . 11  |-  ( b  =  B  ->  (
c  Btwn  <. a ,  b >.  <->  c  Btwn  <. a ,  B >. ) )
1511, 12, 143orbi123d 1334 . . . . . . . . . 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 715 . . . . . . . . 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 2937 . . . . . . . 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 2492 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  e.  ( EE
`  n )  <->  C  e.  ( EE `  n ) ) )
19183anbi3d 1341 . . . . . . . . . 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 4182 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2120breq2d 4429 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
a  Btwn  <. B , 
c >. 
<->  a  Btwn  <. B ,  C >. ) )
22 opeq1 4181 . . . . . . . . . . . 12  |-  ( c  =  C  ->  <. c ,  a >.  =  <. C ,  a >. )
2322breq2d 4429 . . . . . . . . . . 11  |-  ( c  =  C  ->  ( B  Btwn  <. c ,  a
>. 
<->  B  Btwn  <. C , 
a >. ) )
24 breq1 4420 . . . . . . . . . . 11  |-  ( c  =  C  ->  (
c  Btwn  <. a ,  B >.  <->  C  Btwn  <. a ,  B >. ) )
2521, 23, 243orbi123d 1334 . . . . . . . . . 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 715 . . . . . . . . 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 2937 . . . . . . . 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 2492 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
29283anbi1d 1339 . . . . . . . . . 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 4420 . . . . . . . . . . 11  |-  ( a  =  A  ->  (
a  Btwn  <. B ,  C >. 
<->  A  Btwn  <. B ,  C >. ) )
31 opeq2 4182 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. C , 
a >.  =  <. C ,  A >. )
3231breq2d 4429 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( B  Btwn  <. C ,  a
>. 
<->  B  Btwn  <. C ,  A >. ) )
33 opeq1 4181 . . . . . . . . . . . 12  |-  ( a  =  A  ->  <. a ,  B >.  =  <. A ,  B >. )
3433breq2d 4429 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( C  Btwn  <. a ,  B >.  <-> 
C  Btwn  <. A ,  B >. ) )
3530, 32, 343orbi123d 1334 . . . . . . . . . 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 715 . . . . . . . . 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 2937 . . . . . . . 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 1213 . . . . . 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 467 . . . . 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 458 . . . . . . 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 1006 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )  ->  B  e.  ( EE `  N
) )
4342anim2i 571 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  V  /\  B  e.  ( EE `  N )  /\  C  e.  W )
)  ->  ( N  e.  NN  /\  B  e.  ( EE `  N
) ) )
44 3simpa 1002 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  n )  /\  B  e.  ( EE `  n )  /\  C  e.  ( EE `  n
) )  ->  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) )
4544anim2i 571 . . . . . . . . 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 24786 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  B  e.  ( EE `  n
) ) )  ->  N  =  n )
4746adantrrl 728 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N ) )  /\  ( n  e.  NN  /\  ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
) ) ) )  ->  N  =  n )
48 simprrl 772 . . . . . . . . . . 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 2490 . . . . . . . . . . 11  |-  ( N  =  n  ->  ( A  e.  ( EE `  N )  <->  A  e.  ( EE `  n ) ) )
5148, 50syl5ibrcom 225 . . . . . . . . . 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 479 . . . . . . . 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 608 . . . . . . 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 2913 . . . . 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 218 . . . 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 220 . . 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 218 . 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 220 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 187    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1867   E.wrex 2774   _Vcvv 3078   <.cop 3999   class class class wbr 4417   `'ccnv 4844   ` cfv 5592   {coprab 6297   NNcn 10598   EEcee 24761    Btwn cbtwn 24762    Colinear ccolin 30586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-z 10927  df-uz 11149  df-fz 11772  df-ee 24764  df-colinear 30588
This theorem is referenced by:  liness  30694
  Copyright terms: Public domain W3C validator