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Theorem mdegvscale 21674
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 2452 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2452 . . . . . . 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 17646 . . . . . 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 2452 . . . . . . 7  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 eqid 2452 . . . . . . 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 21664 . . . . . 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 6211 . . . . 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 16800 . . . . . . 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 2497 . . . 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 2827 . 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 17649 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
3129, 22, 30syl2anc 661 . . . 4  |-  ( ph  ->  Y  e.  LMod )
321, 29, 22mplsca 17643 . . . . . . 7  |-  ( ph  ->  R  =  (Scalar `  Y ) )
3332fveq2d 5798 . . . . . 6  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
343, 33syl5eq 2505 . . . . 5  |-  ( ph  ->  K  =  ( Base `  (Scalar `  Y )
) )
357, 34eleqtrd 2542 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
36 eqid 2452 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
37 eqid 2452 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
384, 36, 2, 37lmodvscl 17083 . . . 4  |-  ( ( Y  e.  LMod  /\  F  e.  ( Base `  (Scalar `  Y ) )  /\  G  e.  B )  ->  ( F  .x.  G
)  e.  B )
3931, 35, 9, 38syl3anc 1219 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
4014, 1, 4mdegxrcl 21666 . . . 4  |-  ( G  e.  B  ->  ( D `  G )  e.  RR* )
419, 40syl 16 . . 3  |-  ( ph  ->  ( D `  G
)  e.  RR* )
4214, 1, 4, 15, 6, 16mdegleb 21663 . . 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 1370    e. wcel 1758   A.wral 2796   {crab 2800   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4942   "cima 4946   ` cfv 5521  (class class class)co 6195    ^m cmap 7319   Fincfn 7415   RR*cxr 9523    < clt 9524    <_ cle 9525   NNcn 10428   NN0cn0 10685   Basecbs 14287   .rcmulr 14353  Scalarcsca 14355   .scvsca 14356   0gc0g 14492    gsumg cgsu 14493   Ringcrg 16763   LModclmod 17066   mPoly cmpl 17538  ℂfldccnfld 17938   mDeg cmdg 21650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-fzo 11661  df-seq 11919  df-hash 12216  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-0g 14494  df-gsum 14495  df-mnd 15529  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-subg 15792  df-cntz 15949  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-cring 16766  df-lmod 17068  df-lss 17132  df-psr 17541  df-mpl 17543  df-cnfld 17939  df-mdeg 21652
This theorem is referenced by:  deg1vscale  21704
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