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Theorem mdegcl 22635
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 22628 . 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 7896 . . . 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 5336 . . . . . . 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 18349 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
125, 6tdeglem1 22622 . . . . . . . 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 5719 . . . . . . . 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 3500 . . . . . 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 463 . . . . 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 3653 . . . . 5  |-  NN0  C_  ( NN0  u.  { -oo }
)
1816, 17syl6ss 3501 . . . 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 5715 . . . . . . . 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 18278 . . . . . . . . . . 11  |-  Rel  dom mPoly
2322, 2, 3elbasov 14766 . . . . . . . . . 10  |-  ( F  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
2423simprd 461 . . . . . . . . 9  |-  ( F  e.  B  ->  R  e.  _V )
252, 3, 4, 21, 24mplelsfi 18350 . . . . . . . 8  |-  ( F  e.  B  ->  F finSupp  ( 0g `  R ) )
2625fsuppimpd 7828 . . . . . . 7  |-  ( F  e.  B  ->  ( F supp  ( 0g `  R
) )  e.  Fin )
27 imafi 7805 . . . . . . 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 659 . . . . . 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 463 . . . . 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 459 . . . . 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 10795 . . . . . . 7  |-  NN0  C_  RR
32 ressxr 9626 . . . . . . 7  |-  RR  C_  RR*
3331, 32sstri 3498 . . . . . 6  |-  NN0  C_  RR*
3416, 33syl6ss 3501 . . . . 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 11350 . . . . . 6  |-  <  Or  RR*
36 fisupcl 7919 . . . . . 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 668 . . . . 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 1226 . . . 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 3490 . . 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 11518 . . . . 5  |-  sup ( (/)
,  RR* ,  <  )  = -oo
41 ssun2 3654 . . . . . 6  |-  { -oo } 
C_  ( NN0  u.  { -oo } )
42 mnfxr 11326 . . . . . . . 8  |- -oo  e.  RR*
4342elexi 3116 . . . . . . 7  |- -oo  e.  _V
4443snid 4044 . . . . . 6  |- -oo  e.  { -oo }
4541, 44sselii 3486 . . . . 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 2769 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    u. cun 3459    C_ wss 3461   (/)c0 3783   {csn 4016    |-> cmpt 4497    Or wor 4788   `'ccnv 4987   ran crn 4989   "cima 4991   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   supp csupp 6891    ^m cmap 7412   Fincfn 7509   supcsup 7892   RRcr 9480   -oocmnf 9615   RR*cxr 9616    < clt 9617   NNcn 10531   NN0cn0 10791   Basecbs 14716   0gc0g 14929    gsumg cgsu 14930   mPoly cmpl 18197  ℂfldccnfld 18615   mDeg cmdg 22617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-psr 18200  df-mpl 18202  df-cnfld 18616  df-mdeg 22619
This theorem is referenced by:  deg1cl  22649
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