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Theorem mdeg0 19671
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 15556 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 mdeg0.p . . . . 5  |-  P  =  ( I mPoly  R )
32mplgrp 16404 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Grp )  ->  P  e.  Grp )
41, 3sylan2 460 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  P  e.  Grp )
5 eqid 2366 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
6 mdeg0.z . . . 4  |-  .0.  =  ( 0g `  P )
75, 6grpidcl 14720 . . 3  |-  ( P  e.  Grp  ->  .0.  e.  ( Base `  P
) )
8 mdeg0.d . . . 4  |-  D  =  ( I mDeg  R )
9 eqid 2366 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
10 eqid 2366 . . . 4  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  =  { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }
11 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 ) )
128, 2, 5, 9, 10, 11mdegval 19664 . . 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 18 . 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 443 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  V )
151adantl 452 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  R  e.  Grp )
162, 10, 9, 6, 14, 15mpl0 16395 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  .0.  =  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) )
1716cnveqd 4960 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  `'  .0.  =  `' ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) )
1817imaeq1d 5114 . . . . . . 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 5646 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
2019fconst6 5537 . . . . . . . . . 10  |-  ( { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) : { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin } --> _V
2120a1i 10 . . . . . . . . 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 3385 . . . . . . . . . . 11  |-  ( y  e.  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  \  (/) )  ->  y  e.  { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin } )
2319fvconst2 5847 . . . . . . . . . . 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 15 . . . . . . . . . 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 452 . . . . . . . . 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 5765 . . . . . . . 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 3573 . . . . . . . 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 15 . . . . . . 7  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `' ( { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) " ( _V 
\  { ( 0g
`  R ) } ) )  =  (/) )
2918, 28eqtrd 2398 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) )  =  (/) )
3029imaeq2d 5115 . . . . 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 5133 . . . . 5  |-  ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " (/) )  =  (/)
3230, 31syl6eq 2414 . . . 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 7346 . . 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 10795 . . 3  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
3533, 34syl6eq 2414 . 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 2398 1  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( D `  .0.  )  =  -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   {crab 2632   _Vcvv 2873    \ cdif 3235    C_ wss 3238   (/)c0 3543   {csn 3729    e. cmpt 4179    X. cxp 4790   `'ccnv 4791   "cima 4795   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   Fincfn 7006   supcsup 7340    -oocmnf 9012   RR*cxr 9013    < clt 9014   NNcn 9893   NN0cn0 10114   Basecbs 13356   0gc0g 13610    gsumg cgsu 13611   Grpcgrp 14572   Ringcrg 15547   mPoly cmpl 16299  ℂfldccnfld 16593   mDeg cmdg 19654
This theorem is referenced by:  deg1z  19688
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-subg 14828  df-rng 15550  df-psr 16308  df-mpl 16310  df-mdeg 19656
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