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Theorem mdegvscale 22210
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 2467 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2467 . . . . . . 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 465 . . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  G  e.  B )
11 simpr 461 . . . . . . 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 17881 . . . . . 6  |-  ( (
ph  /\  x  e.  { a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } )  ->  ( ( F  .x.  G ) `  x )  =  ( F ( .r `  R ) ( G `
 x ) ) )
1312adantrr 716 . . . . 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 2467 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 eqid 2467 . . . . . . 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 465 . . . . . . 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 755 . . . . . . 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 756 . . . . . . 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 22200 . . . . . 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 6298 . . . . 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 17023 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  K )  ->  ( F ( .r `  R ) ( 0g
`  R ) )  =  ( 0g `  R ) )
2422, 7, 23syl2anc 661 . . . . . 6  |-  ( ph  ->  ( F ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2524adantr 465 . . . . 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 2512 . . . 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 615 . . 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 2878 . 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 17884 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
3129, 22, 30syl2anc 661 . . . 4  |-  ( ph  ->  Y  e.  LMod )
321, 29, 22mplsca 17878 . . . . . . 7  |-  ( ph  ->  R  =  (Scalar `  Y ) )
3332fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
343, 33syl5eq 2520 . . . . 5  |-  ( ph  ->  K  =  ( Base `  (Scalar `  Y )
) )
357, 34eleqtrd 2557 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
36 eqid 2467 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
37 eqid 2467 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
384, 36, 2, 37lmodvscl 17312 . . . 4  |-  ( ( Y  e.  LMod  /\  F  e.  ( Base `  (Scalar `  Y ) )  /\  G  e.  B )  ->  ( F  .x.  G
)  e.  B )
3931, 35, 9, 38syl3anc 1228 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
4014, 1, 4mdegxrcl 22202 . . . 4  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
419, 40syl 16 . . 3  |-  ( ph  ->  ( D `  G
)  e.  RR* )
4214, 1, 4, 15, 6, 16mdegleb 22199 . . 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 661 . 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 232 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  <_  ( D `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   "cima 5002   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   RR*cxr 9623    < clt 9624    <_ cle 9625   NNcn 10532   NN0cn0 10791   Basecbs 14486   .rcmulr 14552  Scalarcsca 14554   .scvsca 14555   0gc0g 14691    gsumg cgsu 14692   Ringcrg 16986   LModclmod 17295   mPoly cmpl 17773  ℂfldccnfld 18191   mDeg cmdg 22186
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-gsum 14694  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-lmod 17297  df-lss 17362  df-psr 17776  df-mpl 17778  df-cnfld 18192  df-mdeg 22188
This theorem is referenced by:  deg1vscale  22240
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