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Theorem mdegleb 19992
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 19991 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
87adantr 453 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( D `  F
)  =  sup (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) , 
RR* ,  <  ) )
98breq1d 4225 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  sup ( ( H "
( `' F "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  <_  G ) )
10 imassrn 5219 . . . 4  |-  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ran  H
112, 3mplrcl 16555 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1211adantr 453 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  I  e.  _V )
135, 6tdeglem1 19986 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1412, 13syl 16 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H : A --> NN0 )
15 frn 5600 . . . . . 6  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
1614, 15syl 16 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  NN0 )
17 nn0ssre 10230 . . . . . 6  |-  NN0  C_  RR
18 ressxr 9134 . . . . . 6  |-  RR  C_  RR*
1917, 18sstri 3359 . . . . 5  |-  NN0  C_  RR*
2016, 19syl6ss 3362 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  RR* )
2110, 20syl5ss 3361 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  C_  RR* )
22 supxrleub 10910 . . 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 650 . 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 5594 . . . . 5  |-  ( H : A --> NN0  ->  H  Fn  A )
2514, 24syl 16 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H  Fn  A )
26 cnvimass 5227 . . . . 5  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
27 eqid 2438 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
28 simpl 445 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  e.  B )
292, 27, 3, 5, 28mplelf 16502 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F : A --> ( Base `  R ) )
30 fdm 5598 . . . . . 6  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
3129, 30syl 16 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  dom  F  =  A )
3226, 31syl5sseq 3398 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( `' F "
( _V  \  {  .0.  } ) )  C_  A )
33 breq1 4218 . . . . 5  |-  ( y  =  ( H `  x )  ->  (
y  <_  G  <->  ( H `  x )  <_  G
) )
3433ralima 5981 . . . 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 644 . . 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 5594 . . . . . . . 8  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3729, 36syl 16 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  Fn  A )
38 elpreima 5853 . . . . . . 7  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " ( _V 
\  {  .0.  }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
3937, 38syl 16 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( x  e.  ( `' F " ( _V 
\  {  .0.  }
) )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
4039imbi1d 310 . . . . 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 435 . . . . . 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 285 . . . . . . . 8  |-  ( ( ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( -.  ( H `  x )  <_  G  ->  -.  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) )
43 simplr 733 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  G  e.  RR* )
4414ffvelrnda 5873 . . . . . . . . . . . 12  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  NN0 )
4519, 44sseldi 3348 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  RR* )
46 xrltnle 9149 . . . . . . . . . . 11  |-  ( ( G  e.  RR*  /\  ( H `  x )  e.  RR* )  ->  ( G  <  ( H `  x )  <->  -.  ( H `  x )  <_  G ) )
4743, 45, 46syl2anc 644 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( G  <  ( H `  x
)  <->  -.  ( H `  x )  <_  G
) )
4847bicomd 194 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( H `  x )  <_  G  <->  G  <  ( H `  x ) ) )
49 ianor 476 . . . . . . . . . . 11  |-  ( -.  ( ( F `  x )  e.  _V  /\  ( F `  x
)  =/=  .0.  )  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
50 eldifsn 3929 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
5149, 50xchnxbir 302 . . . . . . . . . 10  |-  ( -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
52 orcom 378 . . . . . . . . . . . 12  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( -.  ( F `
 x )  =/= 
.0.  \/  -.  ( F `  x )  e.  _V ) )
53 fvex 5745 . . . . . . . . . . . . . 14  |-  ( F `
 x )  e. 
_V
5453notnoti 118 . . . . . . . . . . . . 13  |-  -.  -.  ( F `  x )  e.  _V
5554biorfi 398 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( -.  ( F `  x )  =/=  .0.  \/  -.  ( F `  x )  e.  _V ) )
56 nne 2607 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( F `  x )  =  .0.  )
5752, 55, 563bitr2i 266 . . . . . . . . . . 11  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( F `  x
)  =  .0.  )
5857a1i 11 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( ( -.  ( F `  x
)  e.  _V  \/  -.  ( F `  x
)  =/=  .0.  )  <->  ( F `  x )  =  .0.  ) )
5951, 58syl5bb 250 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( F `  x )  =  .0.  ) )
6048, 59imbi12d 313 . . . . . . . 8  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( ( -.  ( H `  x
)  <_  G  ->  -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  <->  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
6142, 60syl5bb 250 . . . . . . 7  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( (
( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( G  < 
( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
6261pm5.74da 670 . . . . . 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.  ) ) ) )
6341, 62syl5bb 250 . . . . 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.  )
) ) )
6440, 63bitrd 246 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( x  e.  ( `' F "
( _V  \  {  .0.  } ) )  -> 
( H `  x
)  <_  G )  <->  ( x  e.  A  -> 
( G  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )
) ) )
6564ralbidv2 2729 . . 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.  ) ) )
6635, 65bitrd 246 . 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.  ) ) )
679, 23, 663bitrd 272 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   {csn 3816   class class class wbr 4215    e. cmpt 4269   `'ccnv 4880   dom cdm 4881   ran crn 4882   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    ^m cmap 7021   Fincfn 7112   supcsup 7448   RRcr 8994   RR*cxr 9124    < clt 9125    <_ cle 9126   NNcn 10005   NN0cn0 10226   Basecbs 13474   0gc0g 13728    gsumg cgsu 13729   mPoly cmpl 16413  ℂfldccnfld 16708   mDeg cmdg 19981
This theorem is referenced by:  mdeglt  19993  mdegaddle  20002  mdegvscale  20003  mdegle0  20005  mdegmullem  20006  deg1leb  20023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-0g 13732  df-gsum 13733  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-cntz 15121  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-psr 16422  df-mpl 16424  df-cnfld 16709  df-mdeg 19983
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