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Theorem mdegleb 22890
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 22889 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
87adantr 466 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( D `  F
)  =  sup (
( H " ( F supp  .0.  ) ) , 
RR* ,  <  ) )
98breq1d 4436 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  sup ( ( H "
( F supp  .0.  )
) ,  RR* ,  <  )  <_  G ) )
10 imassrn 5199 . . . 4  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
112, 3mplrcl 18648 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1211adantr 466 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  I  e.  _V )
135, 6tdeglem1 22884 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1412, 13syl 17 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H : A --> NN0 )
15 frn 5752 . . . . . 6  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
1614, 15syl 17 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  NN0 )
17 nn0ssre 10873 . . . . . 6  |-  NN0  C_  RR
18 ressxr 9683 . . . . . 6  |-  RR  C_  RR*
1917, 18sstri 3479 . . . . 5  |-  NN0  C_  RR*
2016, 19syl6ss 3482 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  RR* )
2110, 20syl5ss 3481 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( H " ( F supp  .0.  ) )  C_  RR* )
22 supxrleub 11612 . . 3  |-  ( ( ( H " ( F supp  .0.  ) )  C_  RR* 
/\  G  e.  RR* )  ->  ( sup (
( H " ( F supp  .0.  ) ) , 
RR* ,  <  )  <_  G 
<-> 
A. y  e.  ( H " ( F supp 
.0.  ) ) y  <_  G ) )
2321, 22sylancom 671 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( sup ( ( H " ( F supp 
.0.  ) ) , 
RR* ,  <  )  <_  G 
<-> 
A. y  e.  ( H " ( F supp 
.0.  ) ) y  <_  G ) )
24 ffn 5746 . . . . 5  |-  ( H : A --> NN0  ->  H  Fn  A )
2514, 24syl 17 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H  Fn  A )
26 suppssdm 6938 . . . . 5  |-  ( F supp 
.0.  )  C_  dom  F
27 eqid 2429 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
28 simpl 458 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  e.  B )
292, 27, 3, 5, 28mplelf 18592 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F : A --> ( Base `  R ) )
30 fdm 5750 . . . . . 6  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
3129, 30syl 17 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  dom  F  =  A )
3226, 31syl5sseq 3518 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( F supp  .0.  )  C_  A )
33 breq1 4429 . . . . 5  |-  ( y  =  ( H `  x )  ->  (
y  <_  G  <->  ( H `  x )  <_  G
) )
3433ralima 6160 . . . 4  |-  ( ( H  Fn  A  /\  ( F supp  .0.  )  C_  A )  ->  ( A. y  e.  ( H " ( F supp  .0.  ) ) y  <_  G 
<-> 
A. x  e.  ( F supp  .0.  ) ( H `  x )  <_  G ) )
3525, 32, 34syl2anc 665 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( A. y  e.  ( H " ( F supp  .0.  ) ) y  <_  G  <->  A. x  e.  ( F supp  .0.  )
( H `  x
)  <_  G )
)
36 ffn 5746 . . . . . . . 8  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3729, 36syl 17 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  Fn  A )
38 ovex 6333 . . . . . . . . . 10  |-  ( NN0 
^m  I )  e. 
_V
3938rabex 4576 . . . . . . . . 9  |-  { m  e.  ( NN0  ^m  I
)  |  ( `' m " NN )  e.  Fin }  e.  _V
4039a1i 11 . . . . . . . 8  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  { m  e.  ( NN0  ^m  I )  |  ( `' m " NN )  e.  Fin }  e.  _V )
415, 40syl5eqel 2521 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  A  e.  _V )
42 fvex 5891 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
434, 42eqeltri 2513 . . . . . . . 8  |-  .0.  e.  _V
4443a1i 11 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  .0.  e.  _V )
45 elsuppfn 6933 . . . . . . . 8  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
x  e.  ( F supp 
.0.  )  <->  ( x  e.  A  /\  ( F `  x )  =/=  .0.  ) ) )
46 fvex 5891 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4746biantrur 508 . . . . . . . . . . 11  |-  ( ( F `  x )  =/=  .0.  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
48 eldifsn 4128 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
4947, 48bitr4i 255 . . . . . . . . . 10  |-  ( ( F `  x )  =/=  .0.  <->  ( F `  x )  e.  ( _V  \  {  .0.  } ) )
5049a1i 11 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
( F `  x
)  =/=  .0.  <->  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) )
5150anbi2d 708 . . . . . . . 8  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
( x  e.  A  /\  ( F `  x
)  =/=  .0.  )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
5245, 51bitrd 256 . . . . . . 7  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
x  e.  ( F supp 
.0.  )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
5337, 41, 44, 52syl3anc 1264 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( x  e.  ( F supp  .0.  )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
5453imbi1d 318 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( x  e.  ( F supp  .0.  )  ->  ( H `  x
)  <_  G )  <->  ( ( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  ->  ( H `  x )  <_  G ) ) )
55 impexp 447 . . . . . 6  |-  ( ( ( x  e.  A  /\  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  ->  ( H `  x )  <_  G )  <->  ( x  e.  A  ->  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  -> 
( H `  x
)  <_  G )
) )
56 con34b 293 . . . . . . . 8  |-  ( ( ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( -.  ( H `  x )  <_  G  ->  -.  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) )
57 simplr 760 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  G  e.  RR* )
5814ffvelrnda 6037 . . . . . . . . . . . 12  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  NN0 )
5919, 58sseldi 3468 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  RR* )
60 xrltnle 9700 . . . . . . . . . . 11  |-  ( ( G  e.  RR*  /\  ( H `  x )  e.  RR* )  ->  ( G  <  ( H `  x )  <->  -.  ( H `  x )  <_  G ) )
6157, 59, 60syl2anc 665 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( G  <  ( H `  x
)  <->  -.  ( H `  x )  <_  G
) )
6261bicomd 204 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( H `  x )  <_  G  <->  G  <  ( H `  x ) ) )
63 ianor 490 . . . . . . . . . . 11  |-  ( -.  ( ( F `  x )  e.  _V  /\  ( F `  x
)  =/=  .0.  )  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
6463, 48xchnxbir 310 . . . . . . . . . 10  |-  ( -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
65 orcom 388 . . . . . . . . . . . 12  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( -.  ( F `
 x )  =/= 
.0.  \/  -.  ( F `  x )  e.  _V ) )
6646notnoti 126 . . . . . . . . . . . . 13  |-  -.  -.  ( F `  x )  e.  _V
6766biorfi 408 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( -.  ( F `  x )  =/=  .0.  \/  -.  ( F `  x )  e.  _V ) )
68 nne 2631 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( F `  x )  =  .0.  )
6965, 67, 683bitr2i 276 . . . . . . . . . . 11  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( F `  x
)  =  .0.  )
7069a1i 11 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( ( -.  ( F `  x
)  e.  _V  \/  -.  ( F `  x
)  =/=  .0.  )  <->  ( F `  x )  =  .0.  ) )
7164, 70syl5bb 260 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( F `  x )  =  .0.  ) )
7262, 71imbi12d 321 . . . . . . . 8  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( ( -.  ( H `  x
)  <_  G  ->  -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
) )  <->  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
7356, 72syl5bb 260 . . . . . . 7  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( (
( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( G  < 
( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
7473pm5.74da 691 . . . . . 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.  ) ) ) )
7555, 74syl5bb 260 . . . . 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.  )
) ) )
7654, 75bitrd 256 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( x  e.  ( F supp  .0.  )  ->  ( H `  x
)  <_  G )  <->  ( x  e.  A  -> 
( G  <  ( H `  x )  ->  ( F `  x
)  =  .0.  )
) ) )
7776ralbidv2 2867 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( A. x  e.  ( F supp  .0.  )
( H `  x
)  <_  G  <->  A. x  e.  A  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
7835, 77bitrd 256 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( A. y  e.  ( H " ( F supp  .0.  ) ) y  <_  G  <->  A. x  e.  A  ( G  <  ( H `  x
)  ->  ( F `  x )  =  .0.  ) ) )
799, 23, 783bitrd 282 1  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  A. x  e.  A  ( G  <  ( H `
 x )  -> 
( F `  x
)  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   {crab 2786   _Vcvv 3087    \ cdif 3439    C_ wss 3442   {csn 4002   class class class wbr 4426    |-> cmpt 4484   `'ccnv 4853   dom cdm 4854   ran crn 4855   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   supp csupp 6925    ^m cmap 7480   Fincfn 7577   supcsup 7960   RRcr 9537   RR*cxr 9673    < clt 9674    <_ cle 9675   NNcn 10609   NN0cn0 10869   Basecbs 15084   0gc0g 15297    gsumg cgsu 15298   mPoly cmpl 18512  ℂfldccnfld 18905   mDeg cmdg 22879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-0g 15299  df-gsum 15300  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-cntz 16922  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-cring 17718  df-psr 18515  df-mpl 18517  df-cnfld 18906  df-mdeg 22881
This theorem is referenced by:  mdeglt  22891  mdegaddle  22900  mdegvscale  22901  mdegle0  22903  mdegmullem  22904  deg1leb  22921
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