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

Theorem mdegcl 19997
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 2438 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2438 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
6 eqid 2438 . . 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 19991 . 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 7453 . . . 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 2504 . . 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 5219 . . . . . . 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 16555 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
125, 6tdeglem1 19986 . . . . . . . 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 5600 . . . . . . . 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 19 . . . . . . 7  |-  ( F  e.  B  ->  ran  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) )  C_  NN0 )
1510, 14syl5ss 3361 . . . . . 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 453 . . . . 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 3512 . . . . 5  |-  NN0  C_  ( NN0  u.  {  -oo }
)
1816, 17syl6ss 3362 . . . 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 5596 . . . . . . . 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 19 . . . . . . 7  |-  ( F  e.  B  ->  Fun  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) )
21 id 21 . . . . . . . 8  |-  ( F  e.  B  ->  F  e.  B )
22 reldmmpl 16496 . . . . . . . . . 10  |-  Rel  dom mPoly
2322, 2, 3elbasov 13518 . . . . . . . . 9  |-  ( F  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
2423simprd 451 . . . . . . . 8  |-  ( F  e.  B  ->  R  e.  _V )
252, 3, 4, 21, 24mplelsfi 16556 . . . . . . 7  |-  ( F  e.  B  ->  ( `' F " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
26 imafi 7402 . . . . . . 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 644 . . . . . 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 453 . . . . 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 449 . . . . 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 10230 . . . . . . 7  |-  NN0  C_  RR
31 ressxr 9134 . . . . . . 7  |-  RR  C_  RR*
3230, 31sstri 3359 . . . . . 6  |-  NN0  C_  RR*
3316, 32syl6ss 3362 . . . . 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 10739 . . . . . 6  |-  <  Or  RR*
35 fisupcl 7475 . . . . . 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 653 . . . . 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 1185 . . . 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 3351 . . 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 10907 . . . . 5  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
40 ssun2 3513 . . . . . 6  |-  {  -oo } 
C_  ( NN0  u.  { 
-oo } )
41 mnfxr 10719 . . . . . . . 8  |-  -oo  e.  RR*
4241elexi 2967 . . . . . . 7  |-  -oo  e.  _V
4342snid 3843 . . . . . 6  |-  -oo  e.  { 
-oo }
4440, 43sselii 3347 . . . . 5  |-  -oo  e.  ( NN0  u.  {  -oo } )
4539, 44eqeltri 2508 . . . 4  |-  sup ( (/)
,  RR* ,  <  )  e.  ( NN0  u.  {  -oo } )
4645a1i 11 . . 3  |-  ( F  e.  B  ->  sup ( (/) ,  RR* ,  <  )  e.  ( NN0  u.  { 
-oo } ) )
479, 38, 46pm2.61ne 2681 . 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 2512 1  |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u.  {  -oo } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   _Vcvv 2958    \ cdif 3319    u. cun 3320    C_ wss 3322   (/)c0 3630   {csn 3816    e. cmpt 4269    Or wor 4505   `'ccnv 4880   ran crn 4882   "cima 4884   Fun wfun 5451   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   Fincfn 7112   supcsup 7448   RRcr 8994    -oocmnf 9123   RR*cxr 9124    < clt 9125   NNcn 10005   NN0cn0 10226   Basecbs 13474   0gc0g 13728    gsumg cgsu 13729   mPoly cmpl 16413  ℂfldccnfld 16708   mDeg cmdg 19981
This theorem is referenced by:  deg1cl  20011
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  ax-addf 9074  ax-mulf 9075
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-se 4545  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-isom 5466  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-oi 7482  df-card 7831  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-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  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-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-0g 13732  df-gsum 13733  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-cntz 15121  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-psr 16422  df-mpl 16424  df-cnfld 16709  df-mdeg 19983
  Copyright terms: Public domain W3C validator