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Theorem r1pval 22425
Description: Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
r1pval.e  |-  E  =  (rem1p `  R )
r1pval.p  |-  P  =  (Poly1 `  R )
r1pval.b  |-  B  =  ( Base `  P
)
r1pval.q  |-  Q  =  (quot1p `  R )
r1pval.t  |-  .x.  =  ( .r `  P )
r1pval.m  |-  .-  =  ( -g `  P )
Assertion
Ref Expression
r1pval  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F E G )  =  ( F 
.-  ( ( F Q G )  .x.  G ) ) )

Proof of Theorem r1pval
Dummy variables  b 
f  g  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1pval.p . . . . 5  |-  P  =  (Poly1 `  R )
2 r1pval.b . . . . 5  |-  B  =  ( Base `  P
)
31, 2elbasfv 14554 . . . 4  |-  ( F  e.  B  ->  R  e.  _V )
43adantr 465 . . 3  |-  ( ( F  e.  B  /\  G  e.  B )  ->  R  e.  _V )
5 r1pval.e . . . 4  |-  E  =  (rem1p `  R )
6 fveq2 5872 . . . . . . . . . 10  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
76, 1syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
87fveq2d 5876 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
98, 2syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
109csbeq1d 3447 . . . . . 6  |-  ( r  =  R  ->  [_ ( Base `  (Poly1 `  r ) )  /  b ]_ (
f  e.  b ,  g  e.  b  |->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g ) ) )  =  [_ B  /  b ]_ (
f  e.  b ,  g  e.  b  |->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g ) ) ) )
11 fvex 5882 . . . . . . . . 9  |-  ( Base `  P )  e.  _V
122, 11eqeltri 2551 . . . . . . . 8  |-  B  e. 
_V
1312a1i 11 . . . . . . 7  |-  ( r  =  R  ->  B  e.  _V )
14 simpr 461 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  b  =  B )
157fveq2d 5876 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( -g `  (Poly1 `  r ) )  =  ( -g `  P
) )
16 r1pval.m . . . . . . . . . . 11  |-  .-  =  ( -g `  P )
1715, 16syl6eqr 2526 . . . . . . . . . 10  |-  ( r  =  R  ->  ( -g `  (Poly1 `  r ) )  =  .-  )
18 eqidd 2468 . . . . . . . . . 10  |-  ( r  =  R  ->  f  =  f )
197fveq2d 5876 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( .r `  (Poly1 `  r ) )  =  ( .r `  P ) )
20 r1pval.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  P )
2119, 20syl6eqr 2526 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  (Poly1 `  r ) )  =  .x.  )
22 fveq2 5872 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (quot1p `  r )  =  (quot1p `  R ) )
23 r1pval.q . . . . . . . . . . . . 13  |-  Q  =  (quot1p `  R )
2422, 23syl6eqr 2526 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (quot1p `  r )  =  Q )
2524oveqd 6312 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
f (quot1p `  r ) g )  =  ( f Q g ) )
26 eqidd 2468 . . . . . . . . . . 11  |-  ( r  =  R  ->  g  =  g )
2721, 25, 26oveq123d 6316 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g )  =  ( ( f Q g )  .x.  g ) )
2817, 18, 27oveq123d 6316 . . . . . . . . 9  |-  ( r  =  R  ->  (
f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g ) )  =  ( f 
.-  ( ( f Q g )  .x.  g ) ) )
2928adantr 465 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r ) g ) ( .r `  (Poly1 `  r ) ) g ) )  =  ( f  .-  ( ( f Q g ) 
.x.  g ) ) )
3014, 14, 29mpt2eq123dv 6354 . . . . . . 7  |-  ( ( r  =  R  /\  b  =  B )  ->  ( f  e.  b ,  g  e.  b 
|->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r ) g ) ( .r `  (Poly1 `  r ) ) g ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( f  .-  (
( f Q g )  .x.  g ) ) ) )
3113, 30csbied 3467 . . . . . 6  |-  ( r  =  R  ->  [_ B  /  b ]_ (
f  e.  b ,  g  e.  b  |->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g ) ) )  =  ( f  e.  B , 
g  e.  B  |->  ( f  .-  ( ( f Q g ) 
.x.  g ) ) ) )
3210, 31eqtrd 2508 . . . . 5  |-  ( r  =  R  ->  [_ ( Base `  (Poly1 `  r ) )  /  b ]_ (
f  e.  b ,  g  e.  b  |->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g ) ) )  =  ( f  e.  B , 
g  e.  B  |->  ( f  .-  ( ( f Q g ) 
.x.  g ) ) ) )
33 df-r1p 22402 . . . . 5  |- rem1p  =  (
r  e.  _V  |->  [_ ( Base `  (Poly1 `  r
) )  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r ) g ) ( .r `  (Poly1 `  r ) ) g ) ) ) )
3412, 12mpt2ex 6872 . . . . 5  |-  ( f  e.  B ,  g  e.  B  |->  ( f 
.-  ( ( f Q g )  .x.  g ) ) )  e.  _V
3532, 33, 34fvmpt 5957 . . . 4  |-  ( R  e.  _V  ->  (rem1p `  R )  =  ( f  e.  B , 
g  e.  B  |->  ( f  .-  ( ( f Q g ) 
.x.  g ) ) ) )
365, 35syl5eq 2520 . . 3  |-  ( R  e.  _V  ->  E  =  ( f  e.  B ,  g  e.  B  |->  ( f  .-  ( ( f Q g )  .x.  g
) ) ) )
374, 36syl 16 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  E  =  ( f  e.  B ,  g  e.  B  |->  ( f 
.-  ( ( f Q g )  .x.  g ) ) ) )
38 simpl 457 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  f  =  F )
39 oveq12 6304 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f Q g )  =  ( F Q G ) )
40 simpr 461 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
4139, 40oveq12d 6313 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f Q g )  .x.  g
)  =  ( ( F Q G ) 
.x.  G ) )
4238, 41oveq12d 6313 . . 3  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  .-  (
( f Q g )  .x.  g ) )  =  ( F 
.-  ( ( F Q G )  .x.  G ) ) )
4342adantl 466 . 2  |-  ( ( ( F  e.  B  /\  G  e.  B
)  /\  ( f  =  F  /\  g  =  G ) )  -> 
( f  .-  (
( f Q g )  .x.  g ) )  =  ( F 
.-  ( ( F Q G )  .x.  G ) ) )
44 simpl 457 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
45 simpr 461 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  B )
46 ovex 6320 . . 3  |-  ( F 
.-  ( ( F Q G )  .x.  G ) )  e. 
_V
4746a1i 11 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F  .-  (
( F Q G )  .x.  G ) )  e.  _V )
4837, 43, 44, 45, 47ovmpt2d 6425 1  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F E G )  =  ( F 
.-  ( ( F Q G )  .x.  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   [_csb 3440   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Basecbs 14507   .rcmulr 14573   -gcsg 15927  Poly1cpl1 18086  quot1pcq1p 22396  rem1pcr1p 22397
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  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-mpt2 6300  df-1st 6795  df-2nd 6796  df-slot 14511  df-base 14512  df-r1p 22402
This theorem is referenced by:  r1pcl  22426  r1pdeglt  22427  r1pid  22428  dvdsr1p  22430  ig1pdvds  22445
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