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Theorem mdegleb 19466
Description: Property of being of limited degree. (Contributed by Stefan O'Rear, 19-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 ) )
Assertion
Ref Expression
mdegleb  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  A. x  e.  A  ( G  <  ( H `
 x )  -> 
( F `  x
)  =  .0.  )
) )
Distinct variable groups:    A, h    m, I    .0. , h    x, A   
x, B    x, F    x, G    x, H    h, I    x, R    x,  .0.    h, m
Allowed substitution hints:    A( m)    B( h, m)    D( x, h, m)    P( x, h, m)    R( h, m)    F( h, m)    G( h, m)    H( h, m)    I( x)    .0. ( m)

Proof of Theorem mdegleb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mdegval.d . . . . 5  |-  D  =  ( I mDeg  R )
2 mdegval.p . . . . 5  |-  P  =  ( I mPoly  R )
3 mdegval.b . . . . 5  |-  B  =  ( Base `  P
)
4 mdegval.z . . . . 5  |-  .0.  =  ( 0g `  R )
5 mdegval.a . . . . 5  |-  A  =  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }
6 mdegval.h . . . . 5  |-  H  =  ( h  e.  A  |->  (fld 
gsumg  h ) )
71, 2, 3, 4, 5, 6mdegval 19465 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
87adantr 451 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( D `  F
)  =  sup (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) , 
RR* ,  <  ) )
98breq1d 4049 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  sup ( ( H "
( `' F "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  <_  G ) )
10 imassrn 5041 . . . 4  |-  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ran  H
112, 3mplrcl 16247 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1211adantr 451 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  I  e.  _V )
135, 6tdeglem1 19460 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1412, 13syl 15 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H : A --> NN0 )
15 frn 5411 . . . . . 6  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
1614, 15syl 15 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  NN0 )
17 nn0ssre 9985 . . . . . 6  |-  NN0  C_  RR
18 ressxr 8892 . . . . . 6  |-  RR  C_  RR*
1917, 18sstri 3201 . . . . 5  |-  NN0  C_  RR*
2016, 19syl6ss 3204 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  RR* )
2110, 20syl5ss 3203 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  C_  RR* )
22 supxrleub 10661 . . 3  |-  ( ( ( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  C_  RR* 
/\  G  e.  RR* )  ->  ( sup (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) , 
RR* ,  <  )  <_  G 
<-> 
A. y  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) y  <_  G ) )
2321, 22sylancom 648 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( sup ( ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) , 
RR* ,  <  )  <_  G 
<-> 
A. y  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) y  <_  G ) )
24 ffn 5405 . . . . 5  |-  ( H : A --> NN0  ->  H  Fn  A )
2514, 24syl 15 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H  Fn  A )
26 cnvimass 5049 . . . . 5  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
27 eqid 2296 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
28 simpl 443 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  e.  B )
292, 27, 3, 5, 28mplelf 16194 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F : A --> ( Base `  R ) )
30 fdm 5409 . . . . . 6  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
3129, 30syl 15 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  dom  F  =  A )
3226, 31syl5sseq 3239 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( `' F "
( _V  \  {  .0.  } ) )  C_  A )
33 breq1 4042 . . . . 5  |-  ( y  =  ( H `  x )  ->  (
y  <_  G  <->  ( H `  x )  <_  G
) )
3433ralima 5774 . . . 4  |-  ( ( H  Fn  A  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  ( A. y  e.  ( H " ( `' F "
( _V  \  {  .0.  } ) ) ) y  <_  G  <->  A. x  e.  ( `' F "
( _V  \  {  .0.  } ) ) ( H `  x )  <_  G ) )
3525, 32, 34syl2anc 642 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( A. y  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) y  <_  G  <->  A. x  e.  ( `' F "
( _V  \  {  .0.  } ) ) ( H `  x )  <_  G ) )
36 ffn 5405 . . . . . . . 8  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3729, 36syl 15 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  Fn  A )
38 elpreima 5661 . . . . . . 7  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  {  .0.  }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
3937, 38syl 15 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( x  e.  ( `' F " ( _V 
\  {  .0.  }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
4039imbi1d 308 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( x  e.  ( `' F "
( _V  \  {  .0.  } ) )  -> 
( H `  x
)  <_  G )  <->  ( ( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  ->  ( H `  x )  <_  G ) ) )
41 impexp 433 . . . . . 6  |-  ( ( ( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  ->  ( H `  x )  <_  G )  <->  ( x  e.  A  ->  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  -> 
( H `  x
)  <_  G )
) )
42 con34b 283 . . . . . . . 8  |-  ( ( ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( -.  ( H `  x )  <_  G  ->  -.  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) )
43 simplr 731 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  G  e.  RR* )
44 ffvelrn 5679 . . . . . . . . . . . . 13  |-  ( ( H : A --> NN0  /\  x  e.  A )  ->  ( H `  x
)  e.  NN0 )
4514, 44sylan 457 . . . . . . . . . . . 12  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  NN0 )
4619, 45sseldi 3191 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  RR* )
47 xrltnle 8907 . . . . . . . . . . 11  |-  ( ( G  e.  RR*  /\  ( H `  x )  e.  RR* )  ->  ( G  <  ( H `  x )  <->  -.  ( H `  x )  <_  G ) )
4843, 46, 47syl2anc 642 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( G  <  ( H `  x
)  <->  -.  ( H `  x )  <_  G
) )
4948bicomd 192 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( H `  x )  <_  G  <->  G  <  ( H `  x ) ) )
50 ianor 474 . . . . . . . . . . 11  |-  ( -.  ( ( F `  x )  e.  _V  /\  ( F `  x
)  =/=  .0.  )  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
51 eldifsn 3762 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
5250, 51xchnxbir 300 . . . . . . . . . 10  |-  ( -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
53 orcom 376 . . . . . . . . . . . 12  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( -.  ( F `
 x )  =/= 
.0.  \/  -.  ( F `  x )  e.  _V ) )
54 fvex 5555 . . . . . . . . . . . . . 14  |-  ( F `
 x )  e. 
_V
5554notnoti 115 . . . . . . . . . . . . 13  |-  -.  -.  ( F `  x )  e.  _V
5655biorfi 396 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( -.  ( F `  x )  =/=  .0.  \/  -.  ( F `  x )  e.  _V ) )
57 nne 2463 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( F `  x )  =  .0.  )
5853, 56, 573bitr2i 264 . . . . . . . . . . 11  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( F `  x
)  =  .0.  )
5958a1i 10 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( ( -.  ( F `  x
)  e.  _V  \/  -.  ( F `  x
)  =/=  .0.  )  <->  ( F `  x )  =  .0.  ) )
6052, 59syl5bb 248 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( F `  x )  =  .0.  ) )
6149, 60imbi12d 311 . . . . . . . 8  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( ( -.  ( H `  x
)  <_  G  ->  -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  <->  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
6242, 61syl5bb 248 . . . . . . 7  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( (
( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( G  < 
( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
6362pm5.74da 668 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( x  e.  A  ->  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  ->  ( H `  x )  <_  G ) )  <->  ( x  e.  A  ->  ( G  <  ( H `  x )  ->  ( F `  x )  =  .0.  ) ) ) )
6441, 63syl5bb 248 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) )  -> 
( H `  x
)  <_  G )  <->  ( x  e.  A  -> 
( G  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )
) ) )
6540, 64bitrd 244 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( x  e.  ( `' F "
( _V  \  {  .0.  } ) )  -> 
( H `  x
)  <_  G )  <->  ( x  e.  A  -> 
( G  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )
) ) )
6665ralbidv2 2578 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( A. x  e.  ( `' F "
( _V  \  {  .0.  } ) ) ( H `  x )  <_  G  <->  A. x  e.  A  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
6735, 66bitrd 244 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( A. y  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) y  <_  G  <->  A. x  e.  A  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
689, 23, 673bitrd 270 1  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  A. x  e.  A  ( G  <  ( H `
 x )  -> 
( F `  x
)  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -->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:  mdeglt  19467  mdegaddle  19476  mdegvscale  19477  mdegle0  19479  mdegmullem  19480  deg1leb  19497
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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