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

Theorem fvline 24839
Description: Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvline  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  -> 
( ALine B )  =  { x  |  x  Colinear  <. A ,  B >. } )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    N( x)

Proof of Theorem fvline
Dummy variables  a 
b  l  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear
2 fveq2 5541 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32eleq2d 2363 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
42eleq2d 2363 . . . . . . . 8  |-  ( n  =  N  ->  ( B  e.  ( EE `  n )  <->  B  e.  ( EE `  N ) ) )
53, 43anbi12d 1253 . . . . . . 7  |-  ( n  =  N  ->  (
( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
) ) )
65anbi1d 685 . . . . . 6  |-  ( n  =  N  ->  (
( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  )  <->  ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  A  =/=  B )  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
76rspcev 2897 . . . . 5  |-  ( ( N  e.  NN  /\  ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) )  ->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) )
81, 7mpanr2 665 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) )
9 simpr1 961 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  A  e.  ( EE `  N ) )
10 simpr2 962 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  B  e.  ( EE `  N ) )
11 colinearex 24755 . . . . . . . 8  |-  Colinear  e.  _V
1211cnvex 5225 . . . . . . 7  |-  `'  Colinear  e. 
_V
13 ecexg 6680 . . . . . . 7  |-  ( `'  Colinear 
e.  _V  ->  [ <. A ,  B >. ] `'  Colinear  e. 
_V )
1412, 13ax-mp 8 . . . . . 6  |-  [ <. A ,  B >. ] `'  Colinear  e. 
_V
15 eleq1 2356 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
16 neeq1 2467 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  =/=  b  <->  A  =/=  b ) )
1715, 163anbi13d 1254 . . . . . . . . 9  |-  ( a  =  A  ->  (
( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  <->  ( A  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  A  =/=  b
) ) )
18 opeq1 3812 . . . . . . . . . . 11  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
19 eceq1 6712 . . . . . . . . . . 11  |-  ( <.
a ,  b >.  =  <. A ,  b
>.  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2018, 19syl 15 . . . . . . . . . 10  |-  ( a  =  A  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2120eqeq2d 2307 . . . . . . . . 9  |-  ( a  =  A  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  b
>. ] `'  Colinear  ) )
2217, 21anbi12d 691 . . . . . . . 8  |-  ( a  =  A  ->  (
( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  ( ( A  e.  ( EE `  n )  /\  b  e.  ( EE `  n
)  /\  A  =/=  b )  /\  l  =  [ <. A ,  b
>. ] `'  Colinear  ) ) )
2322rexbidv 2577 . . . . . . 7  |-  ( a  =  A  ->  ( E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  )  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  b  e.  ( EE `  n
)  /\  A  =/=  b )  /\  l  =  [ <. A ,  b
>. ] `'  Colinear  ) ) )
24 eleq1 2356 . . . . . . . . . 10  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
25 neeq2 2468 . . . . . . . . . 10  |-  ( b  =  B  ->  ( A  =/=  b  <->  A  =/=  B ) )
2624, 253anbi23d 1255 . . . . . . . . 9  |-  ( b  =  B  ->  (
( A  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  A  =/=  b
)  <->  ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
) ) )
27 opeq2 3813 . . . . . . . . . . 11  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
28 eceq1 6712 . . . . . . . . . . 11  |-  ( <. A ,  b >.  = 
<. A ,  B >.  ->  [ <. A ,  b
>. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  )
2927, 28syl 15 . . . . . . . . . 10  |-  ( b  =  B  ->  [ <. A ,  b >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  )
3029eqeq2d 2307 . . . . . . . . 9  |-  ( b  =  B  ->  (
l  =  [ <. A ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  B >. ] `'  Colinear  ) )
3126, 30anbi12d 691 . . . . . . . 8  |-  ( b  =  B  ->  (
( ( A  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  A  =/=  b
)  /\  l  =  [ <. A ,  b
>. ] `'  Colinear  )  <->  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  A  =/=  B )  /\  l  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
3231rexbidv 2577 . . . . . . 7  |-  ( b  =  B  ->  ( E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  A  =/=  b
)  /\  l  =  [ <. A ,  b
>. ] `'  Colinear  )  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  A  =/=  B )  /\  l  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
33 eqeq1 2302 . . . . . . . . 9  |-  ( l  =  [ <. A ,  B >. ] `'  Colinear  -> 
( l  =  [ <. A ,  B >. ] `' 
Colinear  <->  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  ) )
3433anbi2d 684 . . . . . . . 8  |-  ( l  =  [ <. A ,  B >. ] `'  Colinear  -> 
( ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  A  =/=  B )  /\  l  =  [ <. A ,  B >. ] `'  Colinear  )  <->  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  A  =/=  B )  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
3534rexbidv 2577 . . . . . . 7  |-  ( l  =  [ <. A ,  B >. ] `'  Colinear  -> 
( E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  A  =/=  B )  /\  l  =  [ <. A ,  B >. ] `'  Colinear  )  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n
)  /\  A  =/=  B )  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
3623, 32, 35eloprabg 5951 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  [ <. A ,  B >. ] `' 
Colinear  e.  _V )  -> 
( <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. 
{ <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
3714, 36mp3an3 1266 . . . . 5  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. 
{ <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
389, 10, 37syl2anc 642 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  -> 
( <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. 
{ <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }  <->  E. n  e.  NN  ( ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  /\  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  ) ) )
398, 38mpbird 223 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. 
{ <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) } )
40 df-ov 5877 . . . 4  |-  ( ALine B )  =  (Line `  <. A ,  B >. )
41 df-br 4040 . . . . . 6  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. Line
)
42 df-line2 24832 . . . . . . 7  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
4342eleq2i 2360 . . . . . 6  |-  ( <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. Line  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e.  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) } )
4441, 43bitri 240 . . . . 5  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. 
{ <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) } )
45 funline 24837 . . . . . 6  |-  Fun Line
46 funbrfv 5577 . . . . . 6  |-  ( Fun Line  ->  ( <. A ,  B >.Line [ <. A ,  B >. ] `'  Colinear  ->  (Line ` 
<. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  ) )
4745, 46ax-mp 8 . . . . 5  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  ->  (Line `  <. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  )
4844, 47sylbir 204 . . . 4  |-  ( <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e.  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }  ->  (Line `  <. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  )
4940, 48syl5eq 2340 . . 3  |-  ( <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e.  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }  ->  ( ALine B
)  =  [ <. A ,  B >. ] `'  Colinear  )
5039, 49syl 15 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  -> 
( ALine B )  =  [ <. A ,  B >. ] `'  Colinear  )
51 opex 4253 . . . 4  |-  <. A ,  B >.  e.  _V
52 dfec2 6679 . . . 4  |-  ( <. A ,  B >.  e. 
_V  ->  [ <. A ,  B >. ] `'  Colinear  =  { x  |  <. A ,  B >. `'  Colinear  x } )
5351, 52ax-mp 8 . . 3  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  <. A ,  B >. `'  Colinear  x }
54 vex 2804 . . . . 5  |-  x  e. 
_V
5551, 54brcnv 4880 . . . 4  |-  ( <. A ,  B >. `'  Colinear  x  <->  x  Colinear  <. A ,  B >. )
5655abbii 2408 . . 3  |-  { x  |  <. A ,  B >. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. A ,  B >. }
5753, 56eqtri 2316 . 2  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. A ,  B >. }
5850, 57syl6eq 2344 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  -> 
( ALine B )  =  { x  |  x  Colinear  <. A ,  B >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   _Vcvv 2801   <.cop 3656   class class class wbr 4039   `'ccnv 4704   Fun wfun 5265   ` cfv 5271  (class class class)co 5874   {coprab 5875   [cec 6674   NNcn 9762   EEcee 24588    Colinear ccolin 24732  Linecline2 24829
This theorem is referenced by:  liness  24840  fvline2  24841  ellines  24847
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-rep 4147  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-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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-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-recs 6404  df-rdg 6439  df-ec 6678  df-nn 9763  df-colinear 24736  df-line2 24832
  Copyright terms: Public domain W3C validator