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Theorem mdegvscale 19461
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 2283 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2283 . . . . . . 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 16192 . . . . . 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 2283 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 eqid 2283 . . . . . . 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 19451 . . . . . 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 5874 . . . . 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 15378 . . . . . . 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 2319 . . . 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 2626 . 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 16195 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
3129, 22, 30syl2anc 642 . . . 4  |-  ( ph  ->  Y  e.  LMod )
321, 29, 22mplsca 16189 . . . . . . 7  |-  ( ph  ->  R  =  (Scalar `  Y ) )
3332fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
343, 33syl5eq 2327 . . . . 5  |-  ( ph  ->  K  =  ( Base `  (Scalar `  Y )
) )
357, 34eleqtrd 2359 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
36 eqid 2283 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
37 eqid 2283 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
384, 36, 2, 37lmodvscl 15644 . . . 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 19453 . . . 4  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
419, 40syl 15 . . 3  |-  ( ph  ->  ( D `  G
)  e.  RR* )
4214, 1, 4, 15, 6, 16mdegleb 19450 . . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400    gsumg cgsu 13401   Ringcrg 15337   LModclmod 15627   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  deg1vscale  19490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-lmod 15629  df-lss 15690  df-psr 16098  df-mpl 16100  df-cnfld 16378  df-mdeg 19441
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