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Theorem mdeg0 19950
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 15628 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 mdeg0.p . . . . 5  |-  P  =  ( I mPoly  R )
32mplgrp 16472 . . . 4  |-  ( ( I  e.  V  /\  R  e.  Grp )  ->  P  e.  Grp )
41, 3sylan2 461 . . 3  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  P  e.  Grp )
5 eqid 2408 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
6 mdeg0.z . . . 4  |-  .0.  =  ( 0g `  P )
75, 6grpidcl 14792 . . 3  |-  ( P  e.  Grp  ->  .0.  e.  ( Base `  P
) )
8 mdeg0.d . . . 4  |-  D  =  ( I mDeg  R )
9 eqid 2408 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
10 eqid 2408 . . . 4  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  =  { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }
11 eqid 2408 . . . 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 19943 . . 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 444 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  I  e.  V )
151adantl 453 . . . . . . . . . 10  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  R  e.  Grp )
162, 10, 9, 6, 14, 15mpl0 16463 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  .0.  =  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) )
1716cnveqd 5011 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  `'  .0.  =  `' ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) )
1817imaeq1d 5165 . . . . . . 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 5705 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
2019fconst6 5596 . . . . . . . . . 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 3433 . . . . . . . . . . 11  |-  ( y  e.  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  \  (/) )  ->  y  e.  { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin } )
2319fvconst2 5910 . . . . . . . . . . 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 453 . . . . . . . . 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 5826 . . . . . . . 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 3622 . . . . . . . 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 2440 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) )  =  (/) )
3029imaeq2d 5166 . . . . 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 5184 . . . . 5  |-  ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " (/) )  =  (/)
3230, 31syl6eq 2456 . . . 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 7413 . . 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 10862 . . 3  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
3533, 34syl6eq 2456 . 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 2440 1  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( D `  .0.  )  =  -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674   _Vcvv 2920    \ cdif 3281    C_ wss 3284   (/)c0 3592   {csn 3778    e. cmpt 4230    X. cxp 4839   `'ccnv 4840   "cima 4844   -->wf 5413   ` cfv 5417  (class class class)co 6044    ^m cmap 6981   Fincfn 7072   supcsup 7407    -oocmnf 9078   RR*cxr 9079    < clt 9080   NNcn 9960   NN0cn0 10181   Basecbs 13428   0gc0g 13682    gsumg cgsu 13683   Grpcgrp 14644   Ringcrg 15619   mPoly cmpl 16367  ℂfldccnfld 16662   mDeg cmdg 19933
This theorem is referenced by:  deg1z  19967
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-tset 13507  df-0g 13686  df-mnd 14649  df-grp 14771  df-minusg 14772  df-subg 14900  df-rng 15622  df-psr 16376  df-mpl 16378  df-mdeg 19935
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