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

Theorem mdegcl 19455
Description: Sharp closure for multivariate polynomials. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Hypotheses
Ref Expression
mdegcl.d  |-  D  =  ( I mDeg  R )
mdegcl.p  |-  P  =  ( I mPoly  R )
mdegcl.b  |-  B  =  ( Base `  P
)
Assertion
Ref Expression
mdegcl  |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u.  {  -oo } ) )

Proof of Theorem mdegcl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdegcl.d . . 3  |-  D  =  ( I mDeg  R )
2 mdegcl.p . . 3  |-  P  =  ( I mPoly  R )
3 mdegcl.b . . 3  |-  B  =  ( Base `  P
)
4 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2283 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
6 eqid 2283 . . 3  |-  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )  =  ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) )
71, 2, 3, 4, 5, 6mdegval 19449 . 2  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  ) )
8 supeq1 7198 . . . 4  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =  (/)  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  =  sup ( (/) ,  RR* ,  <  ) )
98eleq1d 2349 . . 3  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =  (/)  ->  ( sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  e.  ( NN0  u.  {  -oo } )  <->  sup ( (/)
,  RR* ,  <  )  e.  ( NN0  u.  {  -oo } ) ) )
10 imassrn 5025 . . . . . . 7  |-  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) )  C_  ran  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) )
112, 3mplrcl 16231 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
125, 6tdeglem1 19444 . . . . . . . 8  |-  ( I  e.  _V  ->  (
b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> NN0 )
13 frn 5395 . . . . . . . 8  |-  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> NN0  ->  ran  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) )  C_  NN0 )
1411, 12, 133syl 18 . . . . . . 7  |-  ( F  e.  B  ->  ran  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) )  C_  NN0 )
1510, 14syl5ss 3190 . . . . . 6  |-  ( F  e.  B  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  C_  NN0 )
1615adantr 451 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  -> 
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  C_  NN0 )
17 ssun1 3338 . . . . 5  |-  NN0  C_  ( NN0  u.  {  -oo }
)
1816, 17syl6ss 3191 . . . 4  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  -> 
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  C_  ( NN0  u. 
{  -oo } ) )
19 ffun 5391 . . . . . . . 8  |-  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) : { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin } --> NN0  ->  Fun  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) )
2011, 12, 193syl 18 . . . . . . 7  |-  ( F  e.  B  ->  Fun  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) )
21 id 19 . . . . . . . 8  |-  ( F  e.  B  ->  F  e.  B )
22 reldmmpl 16172 . . . . . . . . . 10  |-  Rel  dom mPoly
2322, 2, 3elbasov 13192 . . . . . . . . 9  |-  ( F  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
2423simprd 449 . . . . . . . 8  |-  ( F  e.  B  ->  R  e.  _V )
252, 3, 4, 21, 24mplelsfi 16232 . . . . . . 7  |-  ( F  e.  B  ->  ( `' F " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
26 imafi 7148 . . . . . . 7  |-  ( ( Fun  ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) )  /\  ( `' F " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )  ->  ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) )  e. 
Fin )
2720, 25, 26syl2anc 642 . . . . . 6  |-  ( F  e.  B  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  e.  Fin )
2827adantr 451 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  -> 
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  e.  Fin )
29 simpr 447 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  -> 
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )
30 nn0ssre 9969 . . . . . . 7  |-  NN0  C_  RR
31 ressxr 8876 . . . . . . 7  |-  RR  C_  RR*
3230, 31sstri 3188 . . . . . 6  |-  NN0  C_  RR*
3316, 32syl6ss 3191 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  -> 
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  C_  RR* )
34 xrltso 10475 . . . . . 6  |-  <  Or  RR*
35 fisupcl 7218 . . . . . 6  |-  ( (  <  Or  RR*  /\  (
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  e.  Fin  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/)  /\  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  C_  RR* ) )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  e.  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) ) )
3634, 35mpan 651 . . . . 5  |-  ( ( ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  e.  Fin  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/)  /\  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  C_  RR* )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  e.  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) ) )
3728, 29, 33, 36syl3anc 1182 . . . 4  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  e.  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) ) )
3818, 37sseldd 3181 . . 3  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( `' F "
( _V  \  {
( 0g `  R
) } ) ) )  =/=  (/) )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  e.  ( NN0  u.  {  -oo } ) )
39 xrsup0 10642 . . . . 5  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
40 ssun2 3339 . . . . . 6  |-  {  -oo } 
C_  ( NN0  u.  { 
-oo } )
41 mnfxr 10456 . . . . . . . 8  |-  -oo  e.  RR*
4241elexi 2797 . . . . . . 7  |-  -oo  e.  _V
4342snid 3667 . . . . . 6  |-  -oo  e.  { 
-oo }
4440, 43sselii 3177 . . . . 5  |-  -oo  e.  ( NN0  u.  {  -oo } )
4539, 44eqeltri 2353 . . . 4  |-  sup ( (/)
,  RR* ,  <  )  e.  ( NN0  u.  {  -oo } )
4645a1i 10 . . 3  |-  ( F  e.  B  ->  sup ( (/) ,  RR* ,  <  )  e.  ( NN0  u.  { 
-oo } ) )
479, 38, 46pm2.61ne 2521 . 2  |-  ( F  e.  B  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( `' F " ( _V 
\  { ( 0g
`  R ) } ) ) ) , 
RR* ,  <  )  e.  ( NN0  u.  {  -oo } ) )
487, 47eqeltrd 2357 1  |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u.  {  -oo } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    Or wor 4313   `'ccnv 4688   ran crn 4690   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   supcsup 7193   RRcr 8736    -oocmnf 8865   RR*cxr 8866    < clt 8867   NNcn 9746   NN0cn0 9965   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  deg1cl  19469
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  ax-addf 8816  ax-mulf 8817
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-se 4353  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-isom 5264  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-oi 7225  df-card 7572  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-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  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-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-psr 16098  df-mpl 16100  df-cnfld 16378  df-mdeg 19441
  Copyright terms: Public domain W3C validator