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Theorem mdeglt 22193
Description: If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.)
Hypotheses
Ref Expression
mdegval.d  |-  D  =  ( I mDeg  R )
mdegval.p  |-  P  =  ( I mPoly  R )
mdegval.b  |-  B  =  ( Base `  P
)
mdegval.z  |-  .0.  =  ( 0g `  R )
mdegval.a  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
mdegval.h  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
mdeglt.f  |-  ( ph  ->  F  e.  B )
medglt.x  |-  ( ph  ->  X  e.  A )
mdeglt.lt  |-  ( ph  ->  ( D `  F
)  <  ( H `  X ) )
Assertion
Ref Expression
mdeglt  |-  ( ph  ->  ( F `  X
)  =  .0.  )
Distinct variable groups:    A, h    m, I    .0. , h    h, I, m
Allowed substitution hints:    ph( h, m)    A( m)    B( h, m)    D( h, m)    P( h, m)    R( h, m)    F( h, m)    H( h, m)    X( h, m)    .0. ( m)

Proof of Theorem mdeglt
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mdeglt.lt . 2  |-  ( ph  ->  ( D `  F
)  <  ( H `  X ) )
2 medglt.x . . 3  |-  ( ph  ->  X  e.  A )
3 mdeglt.f . . . . . . 7  |-  ( ph  ->  F  e.  B )
4 mdegval.d . . . . . . . 8  |-  D  =  ( I mDeg  R )
5 mdegval.p . . . . . . . 8  |-  P  =  ( I mPoly  R )
6 mdegval.b . . . . . . . 8  |-  B  =  ( Base `  P
)
7 mdegval.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
8 mdegval.a . . . . . . . 8  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
9 mdegval.h . . . . . . . 8  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
104, 5, 6, 7, 8, 9mdegval 22190 . . . . . . 7  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
113, 10syl 16 . . . . . 6  |-  ( ph  ->  ( D `  F
)  =  sup (
( H " ( F supp  .0.  ) ) , 
RR* ,  <  ) )
12 imassrn 5339 . . . . . . . 8  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
135, 6mplrcl 17918 . . . . . . . . . 10  |-  ( F  e.  B  ->  I  e.  _V )
148, 9tdeglem1 22184 . . . . . . . . . 10  |-  ( I  e.  _V  ->  H : A --> NN0 )
15 frn 5728 . . . . . . . . . 10  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
163, 13, 14, 154syl 21 . . . . . . . . 9  |-  ( ph  ->  ran  H  C_  NN0 )
17 nn0ssre 10788 . . . . . . . . . 10  |-  NN0  C_  RR
18 ressxr 9626 . . . . . . . . . 10  |-  RR  C_  RR*
1917, 18sstri 3506 . . . . . . . . 9  |-  NN0  C_  RR*
2016, 19syl6ss 3509 . . . . . . . 8  |-  ( ph  ->  ran  H  C_  RR* )
2112, 20syl5ss 3508 . . . . . . 7  |-  ( ph  ->  ( H " ( F supp  .0.  ) )  C_  RR* )
22 supxrcl 11495 . . . . . . 7  |-  ( ( H " ( F supp 
.0.  ) )  C_  RR* 
->  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  )  e.  RR* )
2321, 22syl 16 . . . . . 6  |-  ( ph  ->  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  )  e.  RR* )
2411, 23eqeltrd 2548 . . . . 5  |-  ( ph  ->  ( D `  F
)  e.  RR* )
25 xrleid 11345 . . . . 5  |-  ( ( D `  F )  e.  RR*  ->  ( D `
 F )  <_ 
( D `  F
) )
2624, 25syl 16 . . . 4  |-  ( ph  ->  ( D `  F
)  <_  ( D `  F ) )
274, 5, 6, 7, 8, 9mdegleb 22192 . . . . 5  |-  ( ( F  e.  B  /\  ( D `  F )  e.  RR* )  ->  (
( D `  F
)  <_  ( D `  F )  <->  A. x  e.  A  ( ( D `  F )  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
283, 24, 27syl2anc 661 . . . 4  |-  ( ph  ->  ( ( D `  F )  <_  ( D `  F )  <->  A. x  e.  A  ( ( D `  F
)  <  ( H `  x )  ->  ( F `  x )  =  .0.  ) ) )
2926, 28mpbid 210 . . 3  |-  ( ph  ->  A. x  e.  A  ( ( D `  F )  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )
)
30 fveq2 5857 . . . . . 6  |-  ( x  =  X  ->  ( H `  x )  =  ( H `  X ) )
3130breq2d 4452 . . . . 5  |-  ( x  =  X  ->  (
( D `  F
)  <  ( H `  x )  <->  ( D `  F )  <  ( H `  X )
) )
32 fveq2 5857 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
3332eqeq1d 2462 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  =  .0.  <->  ( F `  X )  =  .0.  ) )
3431, 33imbi12d 320 . . . 4  |-  ( x  =  X  ->  (
( ( D `  F )  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )  <->  ( ( D `  F
)  <  ( H `  X )  ->  ( F `  X )  =  .0.  ) ) )
3534rspcva 3205 . . 3  |-  ( ( X  e.  A  /\  A. x  e.  A  ( ( D `  F
)  <  ( H `  x )  ->  ( F `  x )  =  .0.  ) )  -> 
( ( D `  F )  <  ( H `  X )  ->  ( F `  X
)  =  .0.  )
)
362, 29, 35syl2anc 661 . 2  |-  ( ph  ->  ( ( D `  F )  <  ( H `  X )  ->  ( F `  X
)  =  .0.  )
)
371, 36mpd 15 1  |-  ( ph  ->  ( F `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498   `'ccnv 4991   ran crn 4993   "cima 4995   -->wf 5575   ` cfv 5579  (class class class)co 6275   supp csupp 6891    ^m cmap 7410   Fincfn 7506   supcsup 7889   RRcr 9480   RR*cxr 9616    < clt 9617    <_ cle 9618   NNcn 10525   NN0cn0 10784   Basecbs 14479   0gc0g 14684    gsumg cgsu 14685   mPoly cmpl 17766  ℂfldccnfld 18184   mDeg cmdg 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-0g 14686  df-gsum 14687  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-cntz 16143  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-psr 17769  df-mpl 17771  df-cnfld 18185  df-mdeg 22181
This theorem is referenced by:  mdegaddle  22202  mdegvscale  22203  mdegmullem  22206
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