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Theorem fvline 25785
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 2380 . . . . 5  |-  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear
2 fveq2 5661 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32eleq2d 2447 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
42eleq2d 2447 . . . . . . . 8  |-  ( n  =  N  ->  ( B  e.  ( EE `  n )  <->  B  e.  ( EE `  N ) ) )
53, 43anbi12d 1255 . . . . . . 7  |-  ( n  =  N  ->  (
( A  e.  ( EE `  n )  /\  B  e.  ( EE `  n )  /\  A  =/=  B
)  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
) ) )
65anbi1d 686 . . . . . 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 2988 . . . . 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 666 . . . 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 963 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  A  e.  ( EE `  N ) )
10 simpr2 964 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  B  e.  ( EE `  N ) )
11 colinearex 25701 . . . . . . . 8  |-  Colinear  e.  _V
1211cnvex 5339 . . . . . . 7  |-  `'  Colinear  e. 
_V
13 ecexg 6838 . . . . . . 7  |-  ( `'  Colinear 
e.  _V  ->  [ <. A ,  B >. ] `'  Colinear  e. 
_V )
1412, 13ax-mp 8 . . . . . 6  |-  [ <. A ,  B >. ] `'  Colinear  e. 
_V
15 eleq1 2440 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
16 neeq1 2551 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  =/=  b  <->  A  =/=  b ) )
1715, 163anbi13d 1256 . . . . . . . . 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 3919 . . . . . . . . . . 11  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
19 eceq1 6870 . . . . . . . . . . 11  |-  ( <.
a ,  b >.  =  <. A ,  b
>.  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2018, 19syl 16 . . . . . . . . . 10  |-  ( a  =  A  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2120eqeq2d 2391 . . . . . . . . 9  |-  ( a  =  A  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  b
>. ] `'  Colinear  ) )
2217, 21anbi12d 692 . . . . . . . 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 2663 . . . . . . 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 2440 . . . . . . . . . 10  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
25 neeq2 2552 . . . . . . . . . 10  |-  ( b  =  B  ->  ( A  =/=  b  <->  A  =/=  B ) )
2624, 253anbi23d 1257 . . . . . . . . 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 3920 . . . . . . . . . . 11  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
28 eceq1 6870 . . . . . . . . . . 11  |-  ( <. A ,  b >.  = 
<. A ,  B >.  ->  [ <. A ,  b
>. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  )
2927, 28syl 16 . . . . . . . . . 10  |-  ( b  =  B  ->  [ <. A ,  b >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  )
3029eqeq2d 2391 . . . . . . . . 9  |-  ( b  =  B  ->  (
l  =  [ <. A ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  B >. ] `'  Colinear  ) )
3126, 30anbi12d 692 . . . . . . . 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 2663 . . . . . . 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 2386 . . . . . . . . 9  |-  ( l  =  [ <. A ,  B >. ] `'  Colinear  -> 
( l  =  [ <. A ,  B >. ] `' 
Colinear  <->  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  ) )
3433anbi2d 685 . . . . . . . 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 2663 . . . . . . 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 6093 . . . . . 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 1268 . . . . 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 643 . . . 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 224 . . 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 6016 . . . 4  |-  ( ALine B )  =  (Line `  <. A ,  B >. )
41 df-br 4147 . . . . . 6  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. Line
)
42 df-line2 25778 . . . . . . 7  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
4342eleq2i 2444 . . . . . 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 241 . . . . 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 25783 . . . . . 6  |-  Fun Line
46 funbrfv 5697 . . . . . 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 205 . . . 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 2424 . . 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 16 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  -> 
( ALine B )  =  [ <. A ,  B >. ] `'  Colinear  )
51 opex 4361 . . . 4  |-  <. A ,  B >.  e.  _V
52 dfec2 6837 . . . 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 2895 . . . . 5  |-  x  e. 
_V
5551, 54brcnv 4988 . . . 4  |-  ( <. A ,  B >. `'  Colinear  x  <->  x  Colinear  <. A ,  B >. )
5655abbii 2492 . . 3  |-  { x  |  <. A ,  B >. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. A ,  B >. }
5753, 56eqtri 2400 . 2  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. A ,  B >. }
5850, 57syl6eq 2428 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   E.wrex 2643   _Vcvv 2892   <.cop 3753   class class class wbr 4146   `'ccnv 4810   Fun wfun 5381   ` cfv 5387  (class class class)co 6013   {coprab 6014   [cec 6832   NNcn 9925   EEcee 25534    Colinear ccolin 25678  Linecline2 25775
This theorem is referenced by:  liness  25786  fvline2  25787  ellines  25793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-i2m1 8984  ax-1ne0 8985  ax-rrecex 8988  ax-cnre 8989
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-recs 6562  df-rdg 6597  df-ec 6836  df-nn 9926  df-colinear 25682  df-line2 25778
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