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Theorem q1pval 19555
Description: Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
q1pval.q  |-  Q  =  (quot1p `  R )
q1pval.p  |-  P  =  (Poly1 `  R )
q1pval.b  |-  B  =  ( Base `  P
)
q1pval.d  |-  D  =  ( deg1  `  R )
q1pval.m  |-  .-  =  ( -g `  P )
q1pval.t  |-  .x.  =  ( .r `  P )
Assertion
Ref Expression
q1pval  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F Q G )  =  ( iota_ q  e.  B ( D `
 ( F  .-  ( q  .x.  G
) ) )  < 
( D `  G
) ) )
Distinct variable groups:    B, q    F, q    G, q    P, q    R, q
Allowed substitution hints:    D( q)    Q( q)    .x. ( q)    .- ( q)

Proof of Theorem q1pval
Dummy variables  b 
f  g  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 q1pval.p . . . . 5  |-  P  =  (Poly1 `  R )
2 q1pval.b . . . . 5  |-  B  =  ( Base `  P
)
31, 2elbasfv 13207 . . . 4  |-  ( G  e.  B  ->  R  e.  _V )
4 q1pval.q . . . . 5  |-  Q  =  (quot1p `  R )
5 fveq2 5541 . . . . . . . . 9  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
65, 1syl6eqr 2346 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
76csbeq1d 3100 . . . . . . 7  |-  ( r  =  R  ->  [_ (Poly1 `  r )  /  p ]_ [_ ( Base `  p
)  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) )  = 
[_ P  /  p ]_ [_ ( Base `  p
)  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) ) )
8 fvex 5555 . . . . . . . . . 10  |-  (Poly1 `  R
)  e.  _V
91, 8eqeltri 2366 . . . . . . . . 9  |-  P  e. 
_V
109a1i 10 . . . . . . . 8  |-  ( r  =  R  ->  P  e.  _V )
11 fveq2 5541 . . . . . . . . . . . 12  |-  ( p  =  P  ->  ( Base `  p )  =  ( Base `  P
) )
1211adantl 452 . . . . . . . . . . 11  |-  ( ( r  =  R  /\  p  =  P )  ->  ( Base `  p
)  =  ( Base `  P ) )
1312, 2syl6eqr 2346 . . . . . . . . . 10  |-  ( ( r  =  R  /\  p  =  P )  ->  ( Base `  p
)  =  B )
1413csbeq1d 3100 . . . . . . . . 9  |-  ( ( r  =  R  /\  p  =  P )  ->  [_ ( Base `  p
)  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) )  = 
[_ B  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) ) )
15 fvex 5555 . . . . . . . . . . . 12  |-  ( Base `  P )  e.  _V
162, 15eqeltri 2366 . . . . . . . . . . 11  |-  B  e. 
_V
1716a1i 10 . . . . . . . . . 10  |-  ( ( r  =  R  /\  p  =  P )  ->  B  e.  _V )
18 simpr 447 . . . . . . . . . . 11  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  b  =  B )
19 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
2019ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
21 q1pval.d . . . . . . . . . . . . . . 15  |-  D  =  ( deg1  `  R )
2220, 21syl6eqr 2346 . . . . . . . . . . . . . 14  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( deg1  `  r )  =  D )
23 fveq2 5541 . . . . . . . . . . . . . . . . 17  |-  ( p  =  P  ->  ( -g `  p )  =  ( -g `  P
) )
2423ad2antlr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( -g `  p )  =  ( -g `  P
) )
25 q1pval.m . . . . . . . . . . . . . . . 16  |-  .-  =  ( -g `  P )
2624, 25syl6eqr 2346 . . . . . . . . . . . . . . 15  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( -g `  p )  = 
.-  )
27 eqidd 2297 . . . . . . . . . . . . . . 15  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  f  =  f )
28 fveq2 5541 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  P  ->  ( .r `  p )  =  ( .r `  P
) )
2928ad2antlr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( .r `  p )  =  ( .r `  P
) )
30 q1pval.t . . . . . . . . . . . . . . . . 17  |-  .x.  =  ( .r `  P )
3129, 30syl6eqr 2346 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( .r `  p )  = 
.x.  )
3231oveqd 5891 . . . . . . . . . . . . . . 15  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  (
q ( .r `  p ) g )  =  ( q  .x.  g ) )
3326, 27, 32oveq123d 5895 . . . . . . . . . . . . . 14  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  (
f ( -g `  p
) ( q ( .r `  p ) g ) )  =  ( f  .-  (
q  .x.  g )
) )
3422, 33fveq12d 5547 . . . . . . . . . . . . 13  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  (
( deg1  `
 r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  =  ( D `
 ( f  .-  ( q  .x.  g
) ) ) )
3522fveq1d 5543 . . . . . . . . . . . . 13  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  (
( deg1  `
 r ) `  g )  =  ( D `  g ) )
3634, 35breq12d 4052 . . . . . . . . . . . 12  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  (
( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
)  <->  ( D `  ( f  .-  (
q  .x.  g )
) )  <  ( D `  g )
) )
3718, 36riotaeqbidv 6323 . . . . . . . . . . 11  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) )  =  (
iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) ) )
3818, 18, 37mpt2eq123dv 5926 . . . . . . . . . 10  |-  ( ( ( r  =  R  /\  p  =  P )  /\  b  =  B )  ->  (
f  e.  b ,  g  e.  b  |->  (
iota_ q  e.  b
( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) ) ) )
3917, 38csbied 3136 . . . . . . . . 9  |-  ( ( r  =  R  /\  p  =  P )  ->  [_ B  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) ) ) )
4014, 39eqtrd 2328 . . . . . . . 8  |-  ( ( r  =  R  /\  p  =  P )  ->  [_ ( Base `  p
)  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) ) ) )
4110, 40csbied 3136 . . . . . . 7  |-  ( r  =  R  ->  [_ P  /  p ]_ [_ ( Base `  p )  / 
b ]_ ( f  e.  b ,  g  e.  b  |->  ( iota_ q  e.  b ( ( deg1  `  r
) `  ( f
( -g `  p ) ( q ( .r
`  p ) g ) ) )  < 
( ( deg1  `  r ) `  g ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B ( D `  ( f  .-  (
q  .x.  g )
) )  <  ( D `  g )
) ) )
427, 41eqtrd 2328 . . . . . 6  |-  ( r  =  R  ->  [_ (Poly1 `  r )  /  p ]_ [_ ( Base `  p
)  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) ) ) )
43 df-q1p 19534 . . . . . 6  |- quot1p  =  ( r  e.  _V  |->  [_ (Poly1 `  r )  /  p ]_ [_ ( Base `  p
)  /  b ]_ ( f  e.  b ,  g  e.  b 
|->  ( iota_ q  e.  b ( ( deg1  `  r ) `  ( f ( -g `  p ) ( q ( .r `  p
) g ) ) )  <  ( ( deg1  `  r ) `  g
) ) ) )
4416, 16mpt2ex 6214 . . . . . 6  |-  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B ( D `
 ( f  .-  ( q  .x.  g
) ) )  < 
( D `  g
) ) )  e. 
_V
4542, 43, 44fvmpt 5618 . . . . 5  |-  ( R  e.  _V  ->  (quot1p `  R )  =  ( f  e.  B , 
g  e.  B  |->  (
iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) ) ) )
464, 45syl5eq 2340 . . . 4  |-  ( R  e.  _V  ->  Q  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B ( D `  ( f  .-  (
q  .x.  g )
) )  <  ( D `  g )
) ) )
473, 46syl 15 . . 3  |-  ( G  e.  B  ->  Q  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B ( D `  ( f  .-  (
q  .x.  g )
) )  <  ( D `  g )
) ) )
4847adantl 452 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  Q  =  ( f  e.  B ,  g  e.  B  |->  ( iota_ q  e.  B ( D `
 ( f  .-  ( q  .x.  g
) ) )  < 
( D `  g
) ) ) )
49 id 19 . . . . . . 7  |-  ( f  =  F  ->  f  =  F )
50 oveq2 5882 . . . . . . 7  |-  ( g  =  G  ->  (
q  .x.  g )  =  ( q  .x.  G ) )
5149, 50oveqan12d 5893 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  .-  (
q  .x.  g )
)  =  ( F 
.-  ( q  .x.  G ) ) )
5251fveq2d 5545 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( D `  (
f  .-  ( q  .x.  g ) ) )  =  ( D `  ( F  .-  ( q 
.x.  G ) ) ) )
53 fveq2 5541 . . . . . 6  |-  ( g  =  G  ->  ( D `  g )  =  ( D `  G ) )
5453adantl 452 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( D `  g
)  =  ( D `
 G ) )
5552, 54breq12d 4052 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( D `  ( f  .-  (
q  .x.  g )
) )  <  ( D `  g )  <->  ( D `  ( F 
.-  ( q  .x.  G ) ) )  <  ( D `  G ) ) )
5655riotabidv 6322 . . 3  |-  ( ( f  =  F  /\  g  =  G )  ->  ( iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) )  =  ( iota_ q  e.  B
( D `  ( F  .-  ( q  .x.  G ) ) )  <  ( D `  G ) ) )
5756adantl 452 . 2  |-  ( ( ( F  e.  B  /\  G  e.  B
)  /\  ( f  =  F  /\  g  =  G ) )  -> 
( iota_ q  e.  B
( D `  (
f  .-  ( q  .x.  g ) ) )  <  ( D `  g ) )  =  ( iota_ q  e.  B
( D `  ( F  .-  ( q  .x.  G ) ) )  <  ( D `  G ) ) )
58 simpl 443 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  F  e.  B )
59 simpr 447 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  G  e.  B )
60 riotaex 6324 . . 3  |-  ( iota_ q  e.  B ( D `
 ( F  .-  ( q  .x.  G
) ) )  < 
( D `  G
) )  e.  _V
6160a1i 10 . 2  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( iota_ q  e.  B
( D `  ( F  .-  ( q  .x.  G ) ) )  <  ( D `  G ) )  e. 
_V )
6248, 57, 58, 59, 61ovmpt2d 5991 1  |-  ( ( F  e.  B  /\  G  e.  B )  ->  ( F Q G )  =  ( iota_ q  e.  B ( D `
 ( F  .-  ( q  .x.  G
) ) )  < 
( D `  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313    < clt 8883   Basecbs 13164   .rcmulr 13225   -gcsg 14381  Poly1cpl1 16268   deg1 cdg1 19456  quot1pcq1p 19529
This theorem is referenced by:  q1peqb  19556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-slot 13168  df-base 13169  df-q1p 19534
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