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Theorem fvline 28180
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 2443 . . . . 5  |-  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear
2 fveq2 5696 . . . . . . . . 9  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32eleq2d 2510 . . . . . . . 8  |-  ( n  =  N  ->  ( A  e.  ( EE `  n )  <->  A  e.  ( EE `  N ) ) )
42eleq2d 2510 . . . . . . . 8  |-  ( n  =  N  ->  ( B  e.  ( EE `  n )  <->  B  e.  ( EE `  N ) ) )
53, 43anbi12d 1290 . . . . . . 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 3078 . . . . 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 994 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  A  e.  ( EE `  N ) )
10 simpr2 995 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B ) )  ->  B  e.  ( EE `  N ) )
11 colinearex 28096 . . . . . . . 8  |-  Colinear  e.  _V
1211cnvex 6530 . . . . . . 7  |-  `'  Colinear  e. 
_V
13 ecexg 7110 . . . . . . 7  |-  ( `'  Colinear 
e.  _V  ->  [ <. A ,  B >. ] `'  Colinear  e. 
_V )
1412, 13ax-mp 5 . . . . . 6  |-  [ <. A ,  B >. ] `'  Colinear  e. 
_V
15 eleq1 2503 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  e.  ( EE
`  n )  <->  A  e.  ( EE `  n ) ) )
16 neeq1 2621 . . . . . . . . . 10  |-  ( a  =  A  ->  (
a  =/=  b  <->  A  =/=  b ) )
1715, 163anbi13d 1291 . . . . . . . . 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 4064 . . . . . . . . . . 11  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
19 eceq1 7142 . . . . . . . . . . 11  |-  ( <.
a ,  b >.  =  <. A ,  b
>.  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2018, 19syl 16 . . . . . . . . . 10  |-  ( a  =  A  ->  [ <. a ,  b >. ] `'  Colinear  =  [ <. A ,  b
>. ] `'  Colinear  )
2120eqeq2d 2454 . . . . . . . . 9  |-  ( a  =  A  ->  (
l  =  [ <. a ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  b
>. ] `'  Colinear  ) )
2217, 21anbi12d 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  ) ) )
2322rexbidv 2741 . . . . . . 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 2503 . . . . . . . . . 10  |-  ( b  =  B  ->  (
b  e.  ( EE
`  n )  <->  B  e.  ( EE `  n ) ) )
25 neeq2 2622 . . . . . . . . . 10  |-  ( b  =  B  ->  ( A  =/=  b  <->  A  =/=  B ) )
2624, 253anbi23d 1292 . . . . . . . . 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 4065 . . . . . . . . . . 11  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
28 eceq1 7142 . . . . . . . . . . 11  |-  ( <. A ,  b >.  = 
<. A ,  B >.  ->  [ <. A ,  b
>. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  )
2927, 28syl 16 . . . . . . . . . 10  |-  ( b  =  B  ->  [ <. A ,  b >. ] `'  Colinear  =  [ <. A ,  B >. ] `'  Colinear  )
3029eqeq2d 2454 . . . . . . . . 9  |-  ( b  =  B  ->  (
l  =  [ <. A ,  b >. ] `'  Colinear  <->  l  =  [ <. A ,  B >. ] `'  Colinear  ) )
3126, 30anbi12d 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  ) ) )
3231rexbidv 2741 . . . . . . 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 2449 . . . . . . . . 9  |-  ( l  =  [ <. A ,  B >. ] `'  Colinear  -> 
( l  =  [ <. A ,  B >. ] `' 
Colinear  <->  [ <. A ,  B >. ] `'  Colinear  =  [ <. A ,  B >. ] `' 
Colinear  ) )
3433anbi2d 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  ) ) )
3534rexbidv 2741 . . . . . . 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 6183 . . . . . 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 1303 . . . . 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 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  ) ) )
398, 38mpbird 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  ) } )
40 df-ov 6099 . . . 4  |-  ( ALine B )  =  (Line `  <. A ,  B >. )
41 df-br 4298 . . . . . 6  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  <->  <. <. A ,  B >. ,  [ <. A ,  B >. ] `'  Colinear  >.  e. Line
)
42 df-line2 28173 . . . . . . 7  |- Line  =  { <. <. a ,  b
>. ,  l >.  |  E. n  e.  NN  ( ( a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  l  =  [ <. a ,  b
>. ] `'  Colinear  ) }
4342eleq2i 2507 . . . . . 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 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  ) } )
45 funline 28178 . . . . . 6  |-  Fun Line
46 funbrfv 5735 . . . . . 6  |-  ( Fun Line  ->  ( <. A ,  B >.Line [ <. A ,  B >. ] `'  Colinear  ->  (Line ` 
<. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  ) )
4745, 46ax-mp 5 . . . . 5  |-  ( <. A ,  B >.Line [
<. A ,  B >. ] `' 
Colinear  ->  (Line `  <. A ,  B >. )  =  [ <. A ,  B >. ] `'  Colinear  )
4844, 47sylbir 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  )
4940, 48syl5eq 2487 . . 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 4561 . . . 4  |-  <. A ,  B >.  e.  _V
52 dfec2 7109 . . . 4  |-  ( <. A ,  B >.  e. 
_V  ->  [ <. A ,  B >. ] `'  Colinear  =  { x  |  <. A ,  B >. `'  Colinear  x } )
5351, 52ax-mp 5 . . 3  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  <. A ,  B >. `'  Colinear  x }
54 vex 2980 . . . . 5  |-  x  e. 
_V
5551, 54brcnv 5027 . . . 4  |-  ( <. A ,  B >. `'  Colinear  x  <->  x  Colinear  <. A ,  B >. )
5655abbii 2560 . . 3  |-  { x  |  <. A ,  B >. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. A ,  B >. }
5753, 56eqtri 2463 . 2  |-  [ <. A ,  B >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. A ,  B >. }
5850, 57syl6eq 2491 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 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2611   E.wrex 2721   _Vcvv 2977   <.cop 3888   class class class wbr 4297   `'ccnv 4844   Fun wfun 5417   ` cfv 5423  (class class class)co 6096   {coprab 6097   [cec 7104   NNcn 10327   EEcee 23139    Colinear ccolin 28073  Linecline2 28170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rrecex 9359  ax-cnre 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-om 6482  df-recs 6837  df-rdg 6871  df-ec 7108  df-nn 10328  df-colinear 28075  df-line2 28173
This theorem is referenced by:  liness  28181  fvline2  28182  ellines  28188
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