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Theorem r1pval 20079
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 13512 . . . 4  |-  ( F  e.  B  ->  R  e.  _V )
43adantr 452 . . 3  |-  ( ( F  e.  B  /\  G  e.  B )  ->  R  e.  _V )
5 r1pval.e . . . 4  |-  E  =  (rem1p `  R )
6 fveq2 5728 . . . . . . . . . 10  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
76, 1syl6eqr 2486 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
87fveq2d 5732 . . . . . . . 8  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
98, 2syl6eqr 2486 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
109csbeq1d 3257 . . . . . 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 5742 . . . . . . . . 9  |-  ( Base `  P )  e.  _V
122, 11eqeltri 2506 . . . . . . . 8  |-  B  e. 
_V
1312a1i 11 . . . . . . 7  |-  ( r  =  R  ->  B  e.  _V )
14 simpr 448 . . . . . . . 8  |-  ( ( r  =  R  /\  b  =  B )  ->  b  =  B )
157fveq2d 5732 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( -g `  (Poly1 `  r ) )  =  ( -g `  P
) )
16 r1pval.m . . . . . . . . . . 11  |-  .-  =  ( -g `  P )
1715, 16syl6eqr 2486 . . . . . . . . . 10  |-  ( r  =  R  ->  ( -g `  (Poly1 `  r ) )  =  .-  )
18 eqidd 2437 . . . . . . . . . 10  |-  ( r  =  R  ->  f  =  f )
197fveq2d 5732 . . . . . . . . . . . 12  |-  ( r  =  R  ->  ( .r `  (Poly1 `  r ) )  =  ( .r `  P ) )
20 r1pval.t . . . . . . . . . . . 12  |-  .x.  =  ( .r `  P )
2119, 20syl6eqr 2486 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( .r `  (Poly1 `  r ) )  =  .x.  )
22 fveq2 5728 . . . . . . . . . . . . 13  |-  ( r  =  R  ->  (quot1p `  r )  =  (quot1p `  R ) )
23 r1pval.q . . . . . . . . . . . . 13  |-  Q  =  (quot1p `  R )
2422, 23syl6eqr 2486 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (quot1p `  r )  =  Q )
2524oveqd 6098 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
f (quot1p `  r ) g )  =  ( f Q g ) )
26 eqidd 2437 . . . . . . . . . . 11  |-  ( r  =  R  ->  g  =  g )
2721, 25, 26oveq123d 6102 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g )  =  ( ( f Q g )  .x.  g ) )
2817, 18, 27oveq123d 6102 . . . . . . . . 9  |-  ( r  =  R  ->  (
f ( -g `  (Poly1 `  r ) ) ( ( f (quot1p `  r
) g ) ( .r `  (Poly1 `  r
) ) g ) )  =  ( f 
.-  ( ( f Q g )  .x.  g ) ) )
2928adantr 452 . . . . . . . 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 6136 . . . . . . 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 3293 . . . . . 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 2468 . . . . 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 20056 . . . . 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 6425 . . . . 5  |-  ( f  e.  B ,  g  e.  B  |->  ( f 
.-  ( ( f Q g )  .x.  g ) ) )  e.  _V
3532, 33, 34fvmpt 5806 . . . 4  |-  ( R  e.  _V  ->  (rem1p `  R )  =  ( f  e.  B , 
g  e.  B  |->  ( f  .-  ( ( f Q g ) 
.x.  g ) ) ) )
365, 35syl5eq 2480 . . 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 444 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  f  =  F )
39 oveq12 6090 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f Q g )  =  ( F Q G ) )
40 simpr 448 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
4139, 40oveq12d 6099 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f Q g )  .x.  g
)  =  ( ( F Q G ) 
.x.  G ) )
4238, 41oveq12d 6099 . . 3  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  .-  (
( f Q g )  .x.  g ) )  =  ( F 
.-  ( ( F Q G )  .x.  G ) ) )
4342adantl 453 . 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 444 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
45 simpr 448 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  B )
46 ovex 6106 . . 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 6201 1  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F E G )  =  ( F 
.-  ( ( F Q G )  .x.  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [_csb 3251   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469   .rcmulr 13530   -gcsg 14688  Poly1cpl1 16571  quot1pcq1p 20050  rem1pcr1p 20051
This theorem is referenced by:  r1pcl  20080  r1pdeglt  20081  r1pid  20082  dvdsr1p  20084  ig1pdvds  20099
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-slot 13473  df-base 13474  df-r1p 20056
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