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Theorem mdegvsca 19464
Description: The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a non-zero-divisor. (Contributed by Stefan O'Rear, 28-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 )
mdegvsca.b  |-  B  =  ( Base `  Y
)
mdegvsca.e  |-  E  =  (RLReg `  R )
mdegvsca.p  |-  .x.  =  ( .s `  Y )
mdegvsca.f  |-  ( ph  ->  F  e.  E )
mdegvsca.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
mdegvsca  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( D `  G ) )

Proof of Theorem mdegvsca
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegaddle.y . . . . . . . 8  |-  Y  =  ( I mPoly  R )
2 mdegvsca.p . . . . . . . 8  |-  .x.  =  ( .s `  Y )
3 eqid 2285 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
4 mdegvsca.b . . . . . . . 8  |-  B  =  ( Base `  Y
)
5 eqid 2285 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2285 . . . . . . . 8  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  =  { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }
7 mdegvsca.e . . . . . . . . . 10  |-  E  =  (RLReg `  R )
87, 3rrgss 16035 . . . . . . . . 9  |-  E  C_  ( Base `  R )
9 mdegvsca.f . . . . . . . . 9  |-  ( ph  ->  F  e.  E )
108, 9sseldi 3180 . . . . . . . 8  |-  ( ph  ->  F  e.  ( Base `  R ) )
11 mdegvsca.g . . . . . . . 8  |-  ( ph  ->  G  e.  B )
121, 2, 3, 4, 5, 6, 10, 11mplvsca 16193 . . . . . . 7  |-  ( ph  ->  ( F  .x.  G
)  =  ( ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { F }
)  o F ( .r `  R ) G ) )
1312cnveqd 4859 . . . . . 6  |-  ( ph  ->  `' ( F  .x.  G )  =  `' ( ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { F } )  o F ( .r `  R ) G ) )
1413imaeq1d 5013 . . . . 5  |-  ( ph  ->  ( `' ( F 
.x.  G ) "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' ( ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { F } )  o F ( .r `  R ) G )
" ( _V  \  { ( 0g `  R ) } ) ) )
15 eqid 2285 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 ovex 5885 . . . . . . . 8  |-  ( NN0 
^m  I )  e. 
_V
1716rabex 4167 . . . . . . 7  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  e.  _V
1817a1i 10 . . . . . 6  |-  ( ph  ->  { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  e.  _V )
19 mdegaddle.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
201, 3, 4, 6, 11mplelf 16180 . . . . . 6  |-  ( ph  ->  G : { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin } --> ( Base `  R ) )
217, 3, 5, 15, 18, 19, 9, 20rrgsupp 16034 . . . . 5  |-  ( ph  ->  ( `' ( ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { F }
)  o F ( .r `  R ) G ) " ( _V  \  { ( 0g
`  R ) } ) )  =  ( `' G " ( _V 
\  { ( 0g
`  R ) } ) ) )
2214, 21eqtrd 2317 . . . 4  |-  ( ph  ->  ( `' ( F 
.x.  G ) "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' G " ( _V  \  {
( 0g `  R
) } ) ) )
2322imaeq2d 5014 . . 3  |-  ( ph  ->  ( ( y  e. 
{ x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) "
( `' ( F 
.x.  G ) "
( _V  \  {
( 0g `  R
) } ) ) )  =  ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `' G " ( _V 
\  { ( 0g
`  R ) } ) ) ) )
2423supeq1d 7201 . 2  |-  ( ph  ->  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `' ( F  .x.  G
) " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  =  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `' G " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
25 mdegaddle.i . . . . 5  |-  ( ph  ->  I  e.  V )
261mpllmod 16197 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
2725, 19, 26syl2anc 642 . . . 4  |-  ( ph  ->  Y  e.  LMod )
281, 25, 19mplsca 16191 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  Y ) )
2928fveq2d 5531 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
3010, 29eleqtrd 2361 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
31 eqid 2285 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
32 eqid 2285 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
334, 31, 2, 32lmodvscl 15646 . . . 4  |-  ( ( Y  e.  LMod  /\  F  e.  ( Base `  (Scalar `  Y ) )  /\  G  e.  B )  ->  ( F  .x.  G
)  e.  B )
3427, 30, 11, 33syl3anc 1182 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
35 mdegaddle.d . . . 4  |-  D  =  ( I mDeg  R )
36 eqid 2285 . . . 4  |-  ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) )  =  ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) )
3735, 1, 4, 15, 6, 36mdegval 19451 . . 3  |-  ( ( F  .x.  G )  e.  B  ->  ( D `  ( F  .x.  G ) )  =  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `' ( F  .x.  G
) " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
3834, 37syl 15 . 2  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  sup ( ( ( y  e.  {
x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) "
( `' ( F 
.x.  G ) "
( _V  \  {
( 0g `  R
) } ) ) ) ,  RR* ,  <  ) )
3935, 1, 4, 15, 6, 36mdegval 19451 . . 3  |-  ( G  e.  B  ->  ( D `  G )  =  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `' G " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
4011, 39syl 15 . 2  |-  ( ph  ->  ( D `  G
)  =  sup (
( ( y  e. 
{ x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) "
( `' G "
( _V  \  {
( 0g `  R
) } ) ) ) ,  RR* ,  <  ) )
4124, 38, 403eqtr4d 2327 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( D `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   {crab 2549   _Vcvv 2790    \ cdif 3151   {csn 3642    e. cmpt 4079    X. cxp 4689   `'ccnv 4690   "cima 4694   ` cfv 5257  (class class class)co 5860    o Fcof 6078    ^m cmap 6774   Fincfn 6865   supcsup 7195   RR*cxr 8868    < clt 8869   NNcn 9748   NN0cn0 9967   Basecbs 13150   .rcmulr 13211  Scalarcsca 13213   .scvsca 13214   0gc0g 13402    gsumg cgsu 13403   Ringcrg 15339   LModclmod 15629  RLRegcrlreg 16022   mPoly cmpl 16091  ℂfldccnfld 16379   mDeg cmdg 19441
This theorem is referenced by:  deg1vsca  19493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-tset 13229  df-0g 13406  df-mnd 14369  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-mgp 15328  df-rng 15342  df-ur 15344  df-lmod 15631  df-lss 15692  df-rlreg 16026  df-psr 16100  df-mpl 16102  df-mdeg 19443
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