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Theorem mdegcl 21674
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 2454 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
5 eqid 2454 . . 3  |-  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  =  { a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }
6 eqid 2454 . . 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 21667 . 2  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) ) , 
RR* ,  <  ) )
8 supeq1 7807 . . . 4  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =  (/)  ->  sup (
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) ,  RR* ,  <  )  =  sup ( (/) ,  RR* ,  <  ) )
98eleq1d 2523 . . 3  |-  ( ( ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =  (/)  ->  ( sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) ) , 
RR* ,  <  )  e.  ( NN0  u.  { -oo } )  <->  sup ( (/)
,  RR* ,  <  )  e.  ( NN0  u.  { -oo } ) ) )
10 imassrn 5289 . . . . . . 7  |-  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) )  C_  ran  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) )
112, 3mplrcl 17696 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
125, 6tdeglem1 21661 . . . . . . . 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 5674 . . . . . . . 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 20 . . . . . . 7  |-  ( F  e.  B  ->  ran  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) )  C_  NN0 )
1510, 14syl5ss 3476 . . . . . 6  |-  ( F  e.  B  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) 
C_  NN0 )
1615adantr 465 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) 
C_  NN0 )
17 ssun1 3628 . . . . 5  |-  NN0  C_  ( NN0  u.  { -oo }
)
1816, 17syl6ss 3477 . . . 4  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) 
C_  ( NN0  u.  { -oo } ) )
19 ffun 5670 . . . . . . . 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 20 . . . . . . 7  |-  ( F  e.  B  ->  Fun  ( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) )
21 id 22 . . . . . . . . 9  |-  ( F  e.  B  ->  F  e.  B )
22 reldmmpl 17625 . . . . . . . . . . 11  |-  Rel  dom mPoly
2322, 2, 3elbasov 14341 . . . . . . . . . 10  |-  ( F  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
2423simprd 463 . . . . . . . . 9  |-  ( F  e.  B  ->  R  e.  _V )
252, 3, 4, 21, 24mplelsfi 17697 . . . . . . . 8  |-  ( F  e.  B  ->  F finSupp  ( 0g `  R ) )
2625fsuppimpd 7739 . . . . . . 7  |-  ( F  e.  B  ->  ( F supp  ( 0g `  R
) )  e.  Fin )
27 imafi 7716 . . . . . . 7  |-  ( ( Fun  ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) )  /\  ( F supp  ( 0g `  R ) )  e. 
Fin )  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  e.  Fin )
2820, 26, 27syl2anc 661 . . . . . 6  |-  ( F  e.  B  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  e.  Fin )
2928adantr 465 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  e.  Fin )
30 simpr 461 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )
31 nn0ssre 10695 . . . . . . 7  |-  NN0  C_  RR
32 ressxr 9539 . . . . . . 7  |-  RR  C_  RR*
3331, 32sstri 3474 . . . . . 6  |-  NN0  C_  RR*
3416, 33syl6ss 3477 . . . . 5  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) 
C_  RR* )
35 xrltso 11230 . . . . . 6  |-  <  Or  RR*
36 fisupcl 7829 . . . . . 6  |-  ( (  <  Or  RR*  /\  (
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  e.  Fin  /\  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/)  /\  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) )  C_  RR* ) )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) ) , 
RR* ,  <  )  e.  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) )
3735, 36mpan 670 . . . . 5  |-  ( ( ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  e.  Fin  /\  (
( b  e.  {
a  e.  ( NN0 
^m  I )  |  ( `' a " NN )  e.  Fin } 
|->  (fld 
gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/)  /\  ( ( b  e.  { a  e.  ( NN0  ^m  I )  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) )  C_  RR* )  ->  sup (
( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) ,  RR* ,  <  )  e.  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) )
3829, 30, 34, 37syl3anc 1219 . . . 4  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) ) , 
RR* ,  <  )  e.  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) ) )
3918, 38sseldd 3466 . . 3  |-  ( ( F  e.  B  /\  ( ( b  e. 
{ a  e.  ( NN0  ^m  I )  |  ( `' a
" NN )  e. 
Fin }  |->  (fld  gsumg  b ) ) "
( F supp  ( 0g `  R ) ) )  =/=  (/) )  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) ) , 
RR* ,  <  )  e.  ( NN0  u.  { -oo } ) )
40 xrsup0 11398 . . . . 5  |-  sup ( (/)
,  RR* ,  <  )  = -oo
41 ssun2 3629 . . . . . 6  |-  { -oo } 
C_  ( NN0  u.  { -oo } )
42 mnfxr 11206 . . . . . . . 8  |- -oo  e.  RR*
4342elexi 3088 . . . . . . 7  |- -oo  e.  _V
4443snid 4014 . . . . . 6  |- -oo  e.  { -oo }
4541, 44sselii 3462 . . . . 5  |- -oo  e.  ( NN0  u.  { -oo } )
4640, 45eqeltri 2538 . . . 4  |-  sup ( (/)
,  RR* ,  <  )  e.  ( NN0  u.  { -oo } )
4746a1i 11 . . 3  |-  ( F  e.  B  ->  sup ( (/) ,  RR* ,  <  )  e.  ( NN0  u.  { -oo } ) )
489, 39, 47pm2.61ne 2767 . 2  |-  ( F  e.  B  ->  sup ( ( ( b  e.  { a  e.  ( NN0  ^m  I
)  |  ( `' a " NN )  e.  Fin }  |->  (fld  gsumg  b ) ) " ( F supp  ( 0g `  R
) ) ) , 
RR* ,  <  )  e.  ( NN0  u.  { -oo } ) )
497, 48eqeltrd 2542 1  |-  ( F  e.  B  ->  ( D `  F )  e.  ( NN0  u.  { -oo } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078    u. cun 3435    C_ wss 3437   (/)c0 3746   {csn 3986    |-> cmpt 4459    Or wor 4749   `'ccnv 4948   ran crn 4950   "cima 4952   Fun wfun 5521   -->wf 5523   ` cfv 5527  (class class class)co 6201   supp csupp 6801    ^m cmap 7325   Fincfn 7421   supcsup 7802   RRcr 9393   -oocmnf 9528   RR*cxr 9529    < clt 9530   NNcn 10434   NN0cn0 10691   Basecbs 14293   0gc0g 14498    gsumg cgsu 14499   mPoly cmpl 17544  ℂfldccnfld 17944   mDeg cmdg 21656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-addf 9473  ax-mulf 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-0g 14500  df-gsum 14501  df-mnd 15535  df-submnd 15585  df-grp 15665  df-minusg 15666  df-cntz 15955  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-cring 16772  df-psr 17547  df-mpl 17549  df-cnfld 17945  df-mdeg 21658
This theorem is referenced by:  deg1cl  21688
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