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

Theorem fvline 29721
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 2467 . . . . 5  |-  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear
2 fveq2 5872 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32eleq2d 2537 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
42eleq2d 2537 . . . . . . . 8  |-  ( n  =  N  ->  ( B  e.  ( EE `  n )  <->  B  e.  ( EE `  N ) ) )
53, 43anbi12d 1300 . . . . . . 7  |-  ( n  =  N  ->  (
( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
) ) )
65anbi1d 704 . . . . . 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 3219 . . . . 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 684 . . . 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 1002 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  A  e.  ( EE `  N ) )
10 simpr2 1003 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  B  e.  ( EE `  N ) )
11 colinearex 29637 . . . . . . . 8  |-  Colinear  e.  _V
1211cnvex 6742 . . . . . . 7  |-  `'  Colinear  e. 
_V
13 ecexg 7327 . . . . . . 7  |-  ( `'  Colinear 
e.  _V  ->  [ <. A ,  B >. ] `'  Colinear  e. 
_V )
1412, 13ax-mp 5 . . . . . 6  |-  [ <. A ,  B >. ] `'  Colinear  e. 
_V
15 eleq1 2539 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
16 neeq1 2748 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  =/=  b  <->  A  =/=  b ) )
1715, 163anbi13d 1301 . . . . . . . . 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 4219 . . . . . . . . . . 11  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
1918eceq1d 7360 . . . . . . . . . 10  |-  ( a  =  A  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2019eqeq2d 2481 . . . . . . . . 9  |-  ( a  =  A  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  b
>. ] `'  Colinear  ) )
2117, 20anbi12d 710 . . . . . . . 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  ) ) )
2221rexbidv 2978 . . . . . . 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  ) ) )
23 eleq1 2539 . . . . . . . . . 10  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
24 neeq2 2750 . . . . . . . . . 10  |-  ( b  =  B  ->  ( A  =/=  b  <->  A  =/=  B ) )
2523, 243anbi23d 1302 . . . . . . . . 9  |-  ( b  =  B  ->  (
( A  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  A  =/=  b
)  <->  ( A  e.  ( EE `  n
)  /\  B  e.  ( EE `  n )  /\  A  =/=  B
) ) )
26 opeq2 4220 . . . . . . . . . . 11  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
2726eceq1d 7360 . . . . . . . . . 10  |-  ( b  =  B  ->  [ <. A ,  b >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  )
2827eqeq2d 2481 . . . . . . . . 9  |-  ( b  =  B  ->  (
l  =  [ <. A ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  B >. ] `'  Colinear  ) )
2925, 28anbi12d 710 . . . . . . . 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  ) ) )
3029rexbidv 2978 . . . . . . 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  ) ) )
31 eqeq1 2471 . . . . . . . . 9  |-  ( l  =  [ <. A ,  B >. ] `'  Colinear  -> 
( l  =  [ <. A ,  B >. ] `' 
Colinear  <->  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  ) )
3231anbi2d 703 . . . . . . . 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  ) ) )
3332rexbidv 2978 . . . . . . 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  ) ) )
3422, 30, 33eloprabg 6385 . . . . . 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  ) ) )
3514, 34mp3an3 1313 . . . . 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  ) ) )
369, 10, 35syl2anc 661 . . . 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  ) ) )
378, 36mpbird 232 . . 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  ) } )
38 df-ov 6298 . . . 4  |-  ( ALine B )  =  (Line `  <. A ,  B >. )
39 df-br 4454 . . . . . 6  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. Line
)
40 df-line2 29714 . . . . . . 7  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
4140eleq2i 2545 . . . . . 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  ) } )
4239, 41bitri 249 . . . . 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  ) } )
43 funline 29719 . . . . . 6  |-  Fun Line
44 funbrfv 5912 . . . . . 6  |-  ( Fun Line  ->  ( <. A ,  B >.Line [ <. A ,  B >. ] `'  Colinear  ->  (Line ` 
<. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  ) )
4543, 44ax-mp 5 . . . . 5  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  ->  (Line `  <. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  )
4642, 45sylbir 213 . . . 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  )
4738, 46syl5eq 2520 . . 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  )
4837, 47syl 16 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  -> 
( ALine B )  =  [ <. A ,  B >. ] `'  Colinear  )
49 opex 4717 . . . 4  |-  <. A ,  B >.  e.  _V
50 dfec2 7326 . . . 4  |-  ( <. A ,  B >.  e. 
_V  ->  [ <. A ,  B >. ] `'  Colinear  =  { x  |  <. A ,  B >. `'  Colinear  x } )
5149, 50ax-mp 5 . . 3  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  <. A ,  B >. `'  Colinear  x }
52 vex 3121 . . . . 5  |-  x  e. 
_V
5349, 52brcnv 5191 . . . 4  |-  ( <. A ,  B >. `'  Colinear  x  <->  x  Colinear  <. A ,  B >. )
5453abbii 2601 . . 3  |-  { x  |  <. A ,  B >. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. A ,  B >. }
5551, 54eqtri 2496 . 2  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. A ,  B >. }
5648, 55syl6eq 2524 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2818   _Vcvv 3118   <.cop 4039   class class class wbr 4453   `'ccnv 5004   Fun wfun 5588   ` cfv 5594  (class class class)co 6295   {coprab 6296   [cec 7321   NNcn 10548   EEcee 24014    Colinear ccolin 29614  Linecline2 29711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-ov 6298  df-oprab 6299  df-om 6696  df-recs 7054  df-rdg 7088  df-ec 7325  df-nn 10549  df-colinear 29616  df-line2 29714
This theorem is referenced by:  liness  29722  fvline2  29723  ellines  29729
  Copyright terms: Public domain W3C validator