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Theorem mdeg0 19998
Description: Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.)
Hypotheses
Ref Expression
mdeg0.d  |-  D  =  ( I mDeg  R )
mdeg0.p  |-  P  =  ( I mPoly  R )
mdeg0.z  |-  .0.  =  ( 0g `  P )
Assertion
Ref Expression
mdeg0  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( D `  .0.  )  =  -oo )

Proof of Theorem mdeg0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnggrp 15674 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 mdeg0.p . . . . 5  |-  P  =  ( I mPoly  R )
32mplgrp 16518 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Grp )  ->  P  e.  Grp )
41, 3sylan2 462 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  P  e.  Grp )
5 eqid 2438 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
6 mdeg0.z . . . 4  |-  .0.  =  ( 0g `  P )
75, 6grpidcl 14838 . . 3  |-  ( P  e.  Grp  ->  .0.  e.  ( Base `  P
) )
8 mdeg0.d . . . 4  |-  D  =  ( I mDeg  R )
9 eqid 2438 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
10 eqid 2438 . . . 4  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  =  { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }
11 eqid 2438 . . . 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 ) )
128, 2, 5, 9, 10, 11mdegval 19991 . . 3  |-  (  .0. 
e.  ( Base `  P
)  ->  ( D `  .0.  )  =  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `'  .0.  " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
134, 7, 123syl 19 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( D `  .0.  )  =  sup (
( ( y  e. 
{ x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) "
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
14 simpl 445 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  V )
151adantl 454 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  R  e.  Grp )
162, 10, 9, 6, 14, 15mpl0 16509 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  .0.  =  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) )
1716cnveqd 5051 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  `'  .0.  =  `' ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) )
1817imaeq1d 5205 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) )  =  ( `' ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } )
" ( _V  \  { ( 0g `  R ) } ) ) )
19 fvex 5745 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
2019fconst6 5636 . . . . . . . . . 10  |-  ( { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) : { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin } --> _V
2120a1i 11 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) : { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin } --> _V )
22 eldifi 3471 . . . . . . . . . . 11  |-  ( y  e.  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  \  (/) )  ->  y  e.  { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin } )
2319fvconst2 5950 . . . . . . . . . . 11  |-  ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  ->  ( ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) `
 y )  =  ( 0g `  R
) )
2422, 23syl 16 . . . . . . . . . 10  |-  ( y  e.  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  \  (/) )  ->  ( ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) `  y )  =  ( 0g `  R ) )
2524adantl 454 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  y  e.  ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
\  (/) ) )  -> 
( ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) `
 y )  =  ( 0g `  R
) )
2621, 25suppss 5866 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `' ( { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) " ( _V 
\  { ( 0g
`  R ) } ) )  C_  (/) )
27 ss0 3660 . . . . . . . 8  |-  ( ( `' ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } )
" ( _V  \  { ( 0g `  R ) } ) )  C_  (/)  ->  ( `' ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } )
" ( _V  \  { ( 0g `  R ) } ) )  =  (/) )
2826, 27syl 16 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `' ( { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) " ( _V 
\  { ( 0g
`  R ) } ) )  =  (/) )
2918, 28eqtrd 2470 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) )  =  (/) )
3029imaeq2d 5206 . . . . 5  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( y  e. 
{ x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) "
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) ) )  =  ( ( y  e. 
{ x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) " (/) ) )
31 ima0 5224 . . . . 5  |-  ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " (/) )  =  (/)
3230, 31syl6eq 2486 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( ( y  e. 
{ x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin } 
|->  (fld 
gsumg  y ) ) "
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) ) )  =  (/) )
3332supeq1d 7454 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `'  .0.  " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  =  sup ( (/) ,  RR* ,  <  ) )
34 xrsup0 10907 . . 3  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
3533, 34syl6eq 2486 . 2  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  sup ( ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " ( `'  .0.  " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  = 
-oo )
3613, 35eqtrd 2470 1  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( D `  .0.  )  =  -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   {csn 3816    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   Fincfn 7112   supcsup 7448    -oocmnf 9123   RR*cxr 9124    < clt 9125   NNcn 10005   NN0cn0 10226   Basecbs 13474   0gc0g 13728    gsumg cgsu 13729   Grpcgrp 14690   Ringcrg 15665   mPoly cmpl 16413  ℂfldccnfld 16708   mDeg cmdg 19981
This theorem is referenced by:  deg1z  20015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-subg 14946  df-rng 15668  df-psr 16422  df-mpl 16424  df-mdeg 19983
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