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Theorem mdegvsca 19866
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 2387 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
4 mdegvsca.b . . . . . . . 8  |-  B  =  ( Base `  Y
)
5 eqid 2387 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
6 eqid 2387 . . . . . . . 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 16279 . . . . . . . . 9  |-  E  C_  ( Base `  R )
9 mdegvsca.f . . . . . . . . 9  |-  ( ph  ->  F  e.  E )
108, 9sseldi 3289 . . . . . . . 8  |-  ( ph  ->  F  e.  ( Base `  R ) )
11 mdegvsca.g . . . . . . . 8  |-  ( ph  ->  G  e.  B )
121, 2, 3, 4, 5, 6, 10, 11mplvsca 16437 . . . . . . 7  |-  ( ph  ->  ( F  .x.  G
)  =  ( ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { F }
)  o F ( .r `  R ) G ) )
1312cnveqd 4988 . . . . . 6  |-  ( ph  ->  `' ( F  .x.  G )  =  `' ( ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { F } )  o F ( .r `  R ) G ) )
1413imaeq1d 5142 . . . . 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 2387 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
16 ovex 6045 . . . . . . . 8  |-  ( NN0 
^m  I )  e. 
_V
1716rabex 4295 . . . . . . 7  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  e.  _V
1817a1i 11 . . . . . 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 16424 . . . . . 6  |-  ( ph  ->  G : { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin } --> ( Base `  R ) )
217, 3, 5, 15, 18, 19, 9, 20rrgsupp 16278 . . . . 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 2419 . . . 4  |-  ( ph  ->  ( `' ( F 
.x.  G ) "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' G " ( _V  \  {
( 0g `  R
) } ) ) )
2322imaeq2d 5143 . . 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 7386 . 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 16441 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  Y  e.  LMod )
2725, 19, 26syl2anc 643 . . . 4  |-  ( ph  ->  Y  e.  LMod )
281, 25, 19mplsca 16435 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  Y ) )
2928fveq2d 5672 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  Y )
) )
3010, 29eleqtrd 2463 . . . 4  |-  ( ph  ->  F  e.  ( Base `  (Scalar `  Y )
) )
31 eqid 2387 . . . . 5  |-  (Scalar `  Y )  =  (Scalar `  Y )
32 eqid 2387 . . . . 5  |-  ( Base `  (Scalar `  Y )
)  =  ( Base `  (Scalar `  Y )
)
334, 31, 2, 32lmodvscl 15894 . . . 4  |-  ( ( Y  e.  LMod  /\  F  e.  ( Base `  (Scalar `  Y ) )  /\  G  e.  B )  ->  ( F  .x.  G
)  e.  B )
3427, 30, 11, 33syl3anc 1184 . . 3  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
35 mdegaddle.d . . . 4  |-  D  =  ( I mDeg  R )
36 eqid 2387 . . . 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 19853 . . 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 16 . 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 19853 . . 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 16 . 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 2429 1  |-  ( ph  ->  ( D `  ( F  .x.  G ) )  =  ( D `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899    \ cdif 3260   {csn 3757    e. cmpt 4207    X. cxp 4816   `'ccnv 4817   "cima 4821   ` cfv 5394  (class class class)co 6020    o Fcof 6242    ^m cmap 6954   Fincfn 7045   supcsup 7380   RR*cxr 9052    < clt 9053   NNcn 9932   NN0cn0 10153   Basecbs 13396   .rcmulr 13457  Scalarcsca 13459   .scvsca 13460   0gc0g 13650    gsumg cgsu 13651   Ringcrg 15587   LModclmod 15877  RLRegcrlreg 16266   mPoly cmpl 16335  ℂfldccnfld 16626   mDeg cmdg 19843
This theorem is referenced by:  deg1vsca  19895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-mgp 15576  df-rng 15590  df-ur 15592  df-lmod 15879  df-lss 15936  df-rlreg 16270  df-psr 16344  df-mpl 16346  df-mdeg 19845
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