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Theorem mdeglt 19467
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.)
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 19465 . . . . . . 7  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
113, 10syl 15 . . . . . 6  |-  ( ph  ->  ( D `  F
)  =  sup (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) , 
RR* ,  <  ) )
12 imassrn 5041 . . . . . . . 8  |-  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ran  H
135, 6mplrcl 16247 . . . . . . . . . . 11  |-  ( F  e.  B  ->  I  e.  _V )
143, 13syl 15 . . . . . . . . . 10  |-  ( ph  ->  I  e.  _V )
158, 9tdeglem1 19460 . . . . . . . . . 10  |-  ( I  e.  _V  ->  H : A --> NN0 )
16 frn 5411 . . . . . . . . . 10  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
1714, 15, 163syl 18 . . . . . . . . 9  |-  ( ph  ->  ran  H  C_  NN0 )
18 nn0ssre 9985 . . . . . . . . . 10  |-  NN0  C_  RR
19 ressxr 8892 . . . . . . . . . 10  |-  RR  C_  RR*
2018, 19sstri 3201 . . . . . . . . 9  |-  NN0  C_  RR*
2117, 20syl6ss 3204 . . . . . . . 8  |-  ( ph  ->  ran  H  C_  RR* )
2212, 21syl5ss 3203 . . . . . . 7  |-  ( ph  ->  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  C_  RR* )
23 supxrcl 10649 . . . . . . 7  |-  ( ( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  C_  RR* 
->  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  e.  RR* )
2422, 23syl 15 . . . . . 6  |-  ( ph  ->  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  e.  RR* )
2511, 24eqeltrd 2370 . . . . 5  |-  ( ph  ->  ( D `  F
)  e.  RR* )
26 xrleid 10500 . . . . 5  |-  ( ( D `  F )  e.  RR*  ->  ( D `
 F )  <_ 
( D `  F
) )
2725, 26syl 15 . . . 4  |-  ( ph  ->  ( D `  F
)  <_  ( D `  F ) )
284, 5, 6, 7, 8, 9mdegleb 19466 . . . . 5  |-  ( ( F  e.  B  /\  ( D `  F )  e.  RR* )  ->  (
( D `  F
)  <_  ( D `  F )  <->  A. x  e.  A  ( ( D `  F )  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
293, 25, 28syl2anc 642 . . . 4  |-  ( ph  ->  ( ( D `  F )  <_  ( D `  F )  <->  A. x  e.  A  ( ( D `  F
)  <  ( H `  x )  ->  ( F `  x )  =  .0.  ) ) )
3027, 29mpbid 201 . . 3  |-  ( ph  ->  A. x  e.  A  ( ( D `  F )  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )
)
31 fveq2 5541 . . . . . 6  |-  ( x  =  X  ->  ( H `  x )  =  ( H `  X ) )
3231breq2d 4051 . . . . 5  |-  ( x  =  X  ->  (
( D `  F
)  <  ( H `  x )  <->  ( D `  F )  <  ( H `  X )
) )
33 fveq2 5541 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
3433eqeq1d 2304 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
)  =  .0.  <->  ( F `  X )  =  .0.  ) )
3532, 34imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( ( D `  F )  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )  <->  ( ( D `  F
)  <  ( H `  X )  ->  ( F `  X )  =  .0.  ) ) )
3635rspcva 2895 . . 3  |-  ( ( X  e.  A  /\  A. x  e.  A  ( ( D `  F
)  <  ( H `  x )  ->  ( F `  x )  =  .0.  ) )  -> 
( ( D `  F )  <  ( H `  X )  ->  ( F `  X
)  =  .0.  )
)
372, 30, 36syl2anc 642 . 2  |-  ( ph  ->  ( ( D `  F )  <  ( H `  X )  ->  ( F `  X
)  =  .0.  )
)
381, 37mpd 14 1  |-  ( ph  ->  ( F `  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   ran crn 4706   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   supcsup 7209   RRcr 8752   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   Basecbs 13164   0gc0g 13416    gsumg cgsu 13417   mPoly cmpl 16105  ℂfldccnfld 16393   mDeg cmdg 19455
This theorem is referenced by:  mdegaddle  19476  mdegvscale  19477  mdegmullem  19480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-psr 16114  df-mpl 16116  df-cnfld 16394  df-mdeg 19457
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