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Theorem mdegvsca 19677
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 2366 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
4 mdegvsca.b . . . . . . . 8  |-  B  =  ( Base `  Y
)
5 eqid 2366 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2366 . . . . . . . 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 16243 . . . . . . . . 9  |-  E  C_  ( Base `  R )
9 mdegvsca.f . . . . . . . . 9  |-  ( ph  ->  F  e.  E )
108, 9sseldi 3264 . . . . . . . 8  |-  ( ph  ->  F  e.  ( Base `  R ) )
11 mdegvsca.g . . . . . . . 8  |-  ( ph  ->  G  e.  B )
121, 2, 3, 4, 5, 6, 10, 11mplvsca 16401 . . . . . . 7  |-  ( ph  ->  ( F  .x.  G
)  =  ( ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { F }
)  o F ( .r `  R ) G ) )
1312cnveqd 4960 . . . . . 6  |-  ( ph  ->  `' ( F  .x.  G )  =  `' ( ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { F } )  o F ( .r `  R ) G ) )
1413imaeq1d 5114 . . . . 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 2366 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 ovex 6006 . . . . . . . 8  |-  ( NN0 
^m  I )  e. 
_V
1716rabex 4267 . . . . . . 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 16388 . . . . . 6  |-  ( ph  ->  G : { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin } --> ( Base `  R ) )
217, 3, 5, 15, 18, 19, 9, 20rrgsupp 16242 . . . . 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 2398 . . . 4  |-  ( ph  ->  ( `' ( F 
.x.  G ) "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' G " ( _V  \  {
( 0g `  R
) } ) ) )
2322imaeq2d 5115 . . 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 7346 . 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 16405 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
2725, 19, 26syl2anc 642 . . . 4  |-  ( ph  ->  Y  e.  LMod )
281, 25, 19mplsca 16399 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  Y ) )
2928fveq2d 5636 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
3010, 29eleqtrd 2442 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
31 eqid 2366 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
32 eqid 2366 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
334, 31, 2, 32lmodvscl 15854 . . . 4  |-  ( ( Y  e.  LMod  /\  F  e.  ( Base `  (Scalar `  Y ) )  /\  G  e.  B )  ->  ( F  .x.  G
)  e.  B )
3427, 30, 11, 33syl3anc 1183 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
35 mdegaddle.d . . . 4  |-  D  =  ( I mDeg  R )
36 eqid 2366 . . . 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 19664 . . 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 19664 . . 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 2408 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( D `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   {crab 2632   _Vcvv 2873    \ cdif 3235   {csn 3729    e. cmpt 4179    X. cxp 4790   `'ccnv 4791   "cima 4795   ` cfv 5358  (class class class)co 5981    o Fcof 6203    ^m cmap 6915   Fincfn 7006   supcsup 7340   RR*cxr 9013    < clt 9014   NNcn 9893   NN0cn0 10114   Basecbs 13356   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420   0gc0g 13610    gsumg cgsu 13611   Ringcrg 15547   LModclmod 15837  RLRegcrlreg 16230   mPoly cmpl 16299  ℂfldccnfld 16593   mDeg cmdg 19654
This theorem is referenced by:  deg1vsca  19706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700  df-sbg 14701  df-subg 14828  df-mgp 15536  df-rng 15550  df-ur 15552  df-lmod 15839  df-lss 15900  df-rlreg 16234  df-psr 16308  df-mpl 16310  df-mdeg 19656
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