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Theorem fvline 26070
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 2435 . . . . 5  |-  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear
2 fveq2 5720 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32eleq2d 2502 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
42eleq2d 2502 . . . . . . . 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 3044 . . . . 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 25986 . . . . . . . 8  |-  Colinear  e.  _V
1211cnvex 5398 . . . . . . 7  |-  `'  Colinear  e. 
_V
13 ecexg 6901 . . . . . . 7  |-  ( `'  Colinear 
e.  _V  ->  [ <. A ,  B >. ] `'  Colinear  e. 
_V )
1412, 13ax-mp 8 . . . . . 6  |-  [ <. A ,  B >. ] `'  Colinear  e. 
_V
15 eleq1 2495 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
16 neeq1 2606 . . . . . . . . . 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 3976 . . . . . . . . . . 11  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
19 eceq1 6933 . . . . . . . . . . 11  |-  ( <.
a ,  b >.  =  <. A ,  b
>.  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2018, 19syl 16 . . . . . . . . . 10  |-  ( a  =  A  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2120eqeq2d 2446 . . . . . . . . 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 2718 . . . . . . 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 2495 . . . . . . . . . 10  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
25 neeq2 2607 . . . . . . . . . 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 3977 . . . . . . . . . . 11  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
28 eceq1 6933 . . . . . . . . . . 11  |-  ( <. A ,  b >.  = 
<. A ,  B >.  ->  [ <. A ,  b
>. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  )
2927, 28syl 16 . . . . . . . . . 10  |-  ( b  =  B  ->  [ <. A ,  b >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  )
3029eqeq2d 2446 . . . . . . . . 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 2718 . . . . . . 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 2441 . . . . . . . . 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 2718 . . . . . . 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 6153 . . . . . 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 6076 . . . 4  |-  ( ALine B )  =  (Line `  <. A ,  B >. )
41 df-br 4205 . . . . . 6  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. Line
)
42 df-line2 26063 . . . . . . 7  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
4342eleq2i 2499 . . . . . 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 26068 . . . . . 6  |-  Fun Line
46 funbrfv 5757 . . . . . 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 2479 . . 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 4419 . . . 4  |-  <. A ,  B >.  e.  _V
52 dfec2 6900 . . . 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 2951 . . . . 5  |-  x  e. 
_V
5551, 54brcnv 5047 . . . 4  |-  ( <. A ,  B >. `'  Colinear  x  <->  x  Colinear  <. A ,  B >. )
5655abbii 2547 . . 3  |-  { x  |  <. A ,  B >. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. A ,  B >. }
5753, 56eqtri 2455 . 2  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. A ,  B >. }
5850, 57syl6eq 2483 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 1652    e. wcel 1725   {cab 2421    =/= wne 2598   E.wrex 2698   _Vcvv 2948   <.cop 3809   class class class wbr 4204   `'ccnv 4869   Fun wfun 5440   ` cfv 5446  (class class class)co 6073   {coprab 6074   [cec 6895   NNcn 9992   EEcee 25819    Colinear ccolin 25963  Linecline2 26060
This theorem is referenced by:  liness  26071  fvline2  26072  ellines  26078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-recs 6625  df-rdg 6660  df-ec 6899  df-nn 9993  df-colinear 25967  df-line2 26063
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