MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mdeg0 Unicode version

Theorem mdeg0 19456
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 15346 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 mdeg0.p . . . . 5  |-  P  =  ( I mPoly  R )
32mplgrp 16194 . . . 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 2283 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
6 mdeg0.z . . . 4  |-  .0.  =  ( 0g `  P )
75, 6grpidcl 14510 . . 3  |-  ( P  e.  Grp  ->  .0.  e.  ( Base `  P
) )
8 mdeg0.d . . . 4  |-  D  =  ( I mDeg  R )
9 eqid 2283 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
10 eqid 2283 . . . 4  |-  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  =  { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }
11 eqid 2283 . . . 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 19449 . . 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 16185 . . . . . . . . 9  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  .0.  =  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g `  R ) } ) )
1716cnveqd 4857 . . . . . . . 8  |-  ( ( I  e.  V  /\  R  e.  Ring )  ->  `'  .0.  =  `' ( { x  e.  ( NN0  ^m  I )  |  ( `' x " NN )  e.  Fin }  X.  { ( 0g
`  R ) } ) )
1817imaeq1d 5011 . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( 0g
`  R )  e. 
_V
2019fconst6 5431 . . . . . . . . . 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 3298 . . . . . . . . . . 11  |-  ( y  e.  ( { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  \  (/) )  ->  y  e.  { x  e.  ( NN0 
^m  I )  |  ( `' x " NN )  e.  Fin } )
2319fvconst2 5729 . . . . . . . . . . 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 5658 . . . . . . . 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 3485 . . . . . . . 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 2315 . . . . . 6  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( `'  .0.  " ( _V  \  { ( 0g
`  R ) } ) )  =  (/) )
3029imaeq2d 5012 . . . . 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 5030 . . . . 5  |-  ( ( y  e.  { x  e.  ( NN0  ^m  I
)  |  ( `' x " NN )  e.  Fin }  |->  (fld  gsumg  y ) ) " (/) )  =  (/)
3230, 31syl6eq 2331 . . . 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 7199 . . 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 10642 . . 3  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
3533, 34syl6eq 2331 . 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 2315 1  |-  ( ( I  e.  V  /\  R  e.  Ring )  -> 
( D `  .0.  )  =  -oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   supcsup 7193    -oocmnf 8865   RR*cxr 8866    < clt 8867   NNcn 9746   NN0cn0 9965   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   Grpcgrp 14362   Ringcrg 15337   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  deg1z  19473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-rng 15340  df-psr 16098  df-mpl 16100  df-mdeg 19441
  Copyright terms: Public domain W3C validator