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Theorem mdegcl 19852
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 2380 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2380 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
6 eqid 2380 . . 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 19846 . 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 7378 . . . 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 2446 . . 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 5149 . . . . . . 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 16470 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
125, 6tdeglem1 19841 . . . . . . . 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 5530 . . . . . . . 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 3295 . . . . . 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 452 . . . . 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 3446 . . . . 5  |-  NN0  C_  ( NN0  u.  {  -oo }
)
1816, 17syl6ss 3296 . . . 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 5526 . . . . . . . 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 20 . . . . . . . 8  |-  ( F  e.  B  ->  F  e.  B )
22 reldmmpl 16411 . . . . . . . . . 10  |-  Rel  dom mPoly
2322, 2, 3elbasov 13433 . . . . . . . . 9  |-  ( F  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
2423simprd 450 . . . . . . . 8  |-  ( F  e.  B  ->  R  e.  _V )
252, 3, 4, 21, 24mplelsfi 16471 . . . . . . 7  |-  ( F  e.  B  ->  ( `' F " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin )
26 imafi 7327 . . . . . . 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 643 . . . . . 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 452 . . . . 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 448 . . . . 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 10150 . . . . . . 7  |-  NN0  C_  RR
31 ressxr 9055 . . . . . . 7  |-  RR  C_  RR*
3230, 31sstri 3293 . . . . . 6  |-  NN0  C_  RR*
3316, 32syl6ss 3296 . . . . 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 10659 . . . . . 6  |-  <  Or  RR*
35 fisupcl 7398 . . . . . 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 652 . . . . 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 1184 . . . 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 3285 . . 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 10827 . . . . 5  |-  sup ( (/)
,  RR* ,  <  )  =  -oo
40 ssun2 3447 . . . . . 6  |-  {  -oo } 
C_  ( NN0  u.  { 
-oo } )
41 mnfxr 10639 . . . . . . . 8  |-  -oo  e.  RR*
4241elexi 2901 . . . . . . 7  |-  -oo  e.  _V
4342snid 3777 . . . . . 6  |-  -oo  e.  { 
-oo }
4440, 43sselii 3281 . . . . 5  |-  -oo  e.  ( NN0  u.  {  -oo } )
4539, 44eqeltri 2450 . . . 4  |-  sup ( (/)
,  RR* ,  <  )  e.  ( NN0  u.  {  -oo } )
4645a1i 11 . . 3  |-  ( F  e.  B  ->  sup ( (/) ,  RR* ,  <  )  e.  ( NN0  u.  { 
-oo } ) )
479, 38, 46pm2.61ne 2618 . 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 2454 1  |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u.  {  -oo } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   {crab 2646   _Vcvv 2892    \ cdif 3253    u. cun 3254    C_ wss 3256   (/)c0 3564   {csn 3750    e. cmpt 4200    Or wor 4436   `'ccnv 4810   ran crn 4812   "cima 4814   Fun wfun 5381   -->wf 5383   ` cfv 5387  (class class class)co 6013    ^m cmap 6947   Fincfn 7038   supcsup 7373   RRcr 8915    -oocmnf 9044   RR*cxr 9045    < clt 9046   NNcn 9925   NN0cn0 10146   Basecbs 13389   0gc0g 13643    gsumg cgsu 13644   mPoly cmpl 16328  ℂfldccnfld 16619   mDeg cmdg 19836
This theorem is referenced by:  deg1cl  19866
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-addf 8995  ax-mulf 8996
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-0g 13647  df-gsum 13648  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-cntz 15036  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-cring 15584  df-ur 15585  df-psr 16337  df-mpl 16339  df-cnfld 16620  df-mdeg 19838
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