MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mdegvscale Unicode version

Theorem mdegvscale 19477
Description: The degree of a scalar multiple of a polynomial is at most the degree of the original polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.)
Hypotheses
Ref Expression
mdegaddle.y  |-  Y  =  ( I mPoly  R )
mdegaddle.d  |-  D  =  ( I mDeg  R )
mdegaddle.i  |-  ( ph  ->  I  e.  V )
mdegaddle.r  |-  ( ph  ->  R  e.  Ring )
mdegvscale.b  |-  B  =  ( Base `  Y
)
mdegvscale.k  |-  K  =  ( Base `  R
)
mdegvscale.p  |-  .x.  =  ( .s `  Y )
mdegvscale.f  |-  ( ph  ->  F  e.  K )
mdegvscale.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
mdegvscale  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( D `  G ) )

Proof of Theorem mdegvscale
Dummy variables  x  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegaddle.y . . . . . . 7  |-  Y  =  ( I mPoly  R )
2 mdegvscale.p . . . . . . 7  |-  .x.  =  ( .s `  Y )
3 mdegvscale.k . . . . . . 7  |-  K  =  ( Base `  R
)
4 mdegvscale.b . . . . . . 7  |-  B  =  ( Base `  Y
)
5 eqid 2296 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2296 . . . . . . 7  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
7 mdegvscale.f . . . . . . . 8  |-  ( ph  ->  F  e.  K )
87adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  F  e.  K )
9 mdegvscale.g . . . . . . . 8  |-  ( ph  ->  G  e.  B )
109adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  G  e.  B )
11 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )
121, 2, 3, 4, 5, 6, 8, 10, 11mplvscaval 16208 . . . . . 6  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( ( F  .x.  G ) `  x )  =  ( F ( .r `  R ) ( G `
 x ) ) )
1312adantrr 697 . . . . 5  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  ( ( F 
.x.  G ) `  x )  =  ( F ( .r `  R ) ( G `
 x ) ) )
14 mdegaddle.d . . . . . . 7  |-  D  =  ( I mDeg  R )
15 eqid 2296 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 eqid 2296 . . . . . . 7  |-  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )  =  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )
179adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  G  e.  B
)
18 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  x  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )
19 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  ( D `  G )  <  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x ) )
2014, 1, 4, 15, 6, 16, 17, 18, 19mdeglt 19467 . . . . . 6  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  ( G `  x )  =  ( 0g `  R ) )
2120oveq2d 5890 . . . . 5  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  ( F ( .r `  R ) ( G `  x
) )  =  ( F ( .r `  R ) ( 0g
`  R ) ) )
22 mdegaddle.r . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
233, 5, 15rngrz 15394 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
2422, 7, 23syl2anc 642 . . . . . 6  |-  ( ph  ->  ( F ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2524adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  ( F ( .r `  R ) ( 0g `  R
) )  =  ( 0g `  R ) )
2613, 21, 253eqtrd 2332 . . . 4  |-  ( (
ph  /\  ( x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  /\  ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x ) ) )  ->  ( ( F 
.x.  G ) `  x )  =  ( 0g `  R ) )
2726expr 598 . . 3  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  (
( F  .x.  G
) `  x )  =  ( 0g `  R ) ) )
2827ralrimiva 2639 . 2  |-  ( ph  ->  A. x  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin }  ( ( D `  G )  <  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) `  x )  ->  (
( F  .x.  G
) `  x )  =  ( 0g `  R ) ) )
29 mdegaddle.i . . . . 5  |-  ( ph  ->  I  e.  V )
301mpllmod 16211 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
3129, 22, 30syl2anc 642 . . . 4  |-  ( ph  ->  Y  e.  LMod )
321, 29, 22mplsca 16205 . . . . . . 7  |-  ( ph  ->  R  =  (Scalar `  Y ) )
3332fveq2d 5545 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
343, 33syl5eq 2340 . . . . 5  |-  ( ph  ->  K  =  ( Base `  (Scalar `  Y )
) )
357, 34eleqtrd 2372 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
36 eqid 2296 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
37 eqid 2296 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
384, 36, 2, 37lmodvscl 15660 . . . 4  |-  ( ( Y  e.  LMod  /\  F  e.  ( Base `  (Scalar `  Y ) )  /\  G  e.  B )  ->  ( F  .x.  G
)  e.  B )
3931, 35, 9, 38syl3anc 1182 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
4014, 1, 4mdegxrcl 19469 . . . 4  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
419, 40syl 15 . . 3  |-  ( ph  ->  ( D `  G
)  e.  RR* )
4214, 1, 4, 15, 6, 16mdegleb 19466 . . 3  |-  ( ( ( F  .x.  G
)  e.  B  /\  ( D `  G )  e.  RR* )  ->  (
( D `  ( F  .x.  G ) )  <_  ( D `  G )  <->  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  (
( F  .x.  G
) `  x )  =  ( 0g `  R ) ) ) )
4339, 41, 42syl2anc 642 . 2  |-  ( ph  ->  ( ( D `  ( F  .x.  G ) )  <_  ( D `  G )  <->  A. x  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  ( ( D `  G )  <  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) `  x )  ->  (
( F  .x.  G
) `  x )  =  ( 0g `  R ) ) ) )
4428, 43mpbird 223 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( D `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   0gc0g 13416    gsumg cgsu 13417   Ringcrg 15353   LModclmod 15643   mPoly cmpl 16105  ℂfldccnfld 16393   mDeg cmdg 19455
This theorem is referenced by:  deg1vscale  19506
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-lmod 15645  df-lss 15706  df-psr 16114  df-mpl 16116  df-cnfld 16394  df-mdeg 19457
  Copyright terms: Public domain W3C validator