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Theorem mdegleb 23012
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 23011 . . . 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 4433 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  sup ( ( H "
( F supp  .0.  )
) ,  RR* ,  <  )  <_  G ) )
10 imassrn 5198 . . . 4  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
112, 3mplrcl 18713 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1211adantr 466 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  I  e.  _V )
135, 6tdeglem1 23006 . . . . . . 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 10881 . . . . . 6  |-  NN0  C_  RR
18 ressxr 9692 . . . . . 6  |-  RR  C_  RR*
1917, 18sstri 3473 . . . . 5  |-  NN0  C_  RR*
2016, 19syl6ss 3476 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  RR* )
2110, 20syl5ss 3475 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( H " ( F supp  .0.  ) )  C_  RR* )
22 supxrleub 11620 . . 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 6939 . . . . 5  |-  ( F supp 
.0.  )  C_  dom  F
27 eqid 2422 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
28 simpl 458 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  e.  B )
292, 27, 3, 5, 28mplelf 18657 . . . . . 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 3512 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( F supp  .0.  )  C_  A )
33 breq1 4426 . . . . 5  |-  ( y  =  ( H `  x )  ->  (
y  <_  G  <->  ( H `  x )  <_  G
) )
3433ralima 6161 . . . 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 6334 . . . . . . . . . 10  |-  ( NN0 
^m  I )  e. 
_V
3938rabex 4575 . . . . . . . . 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 2511 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  A  e.  _V )
42 fvex 5892 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
434, 42eqeltri 2503 . . . . . . . 8  |-  .0.  e.  _V
4443a1i 11 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  .0.  e.  _V )
45 elsuppfn 6934 . . . . . . . 8  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
x  e.  ( F supp 
.0.  )  <->  ( x  e.  A  /\  ( F `  x )  =/=  .0.  ) ) )
46 fvex 5892 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4746biantrur 508 . . . . . . . . . . 11  |-  ( ( F `  x )  =/=  .0.  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
48 eldifsn 4125 . . . . . . . . . . 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 6038 . . . . . . . . . . . 12  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  NN0 )
5919, 58sseldi 3462 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  RR* )
60 xrltnle 9709 . . . . . . . . . . 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 2620 . . . . . . . . . . . 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 2857 . . 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 1872    =/= wne 2614   A.wral 2771   {crab 2775   _Vcvv 3080    \ cdif 3433    C_ wss 3436   {csn 3998   class class class wbr 4423    |-> cmpt 4482   `'ccnv 4852   dom cdm 4853   ran crn 4854   "cima 4856    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6306   supp csupp 6926    ^m cmap 7484   Fincfn 7581   supcsup 7964   RRcr 9546   RR*cxr 9682    < clt 9683    <_ cle 9684   NNcn 10617   NN0cn0 10877   Basecbs 15121   0gc0g 15338    gsumg cgsu 15339   mPoly cmpl 18577  ℂfldccnfld 18970   mDeg cmdg 23001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625  ax-addf 9626  ax-mulf 9627
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-er 7375  df-map 7486  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-sup 7966  df-oi 8035  df-card 8382  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-fz 11793  df-fzo 11924  df-seq 12221  df-hash 12523  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-cntz 16971  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-psr 18580  df-mpl 18582  df-cnfld 18971  df-mdeg 23003
This theorem is referenced by:  mdeglt  23013  mdegaddle  23022  mdegvscale  23023  mdegle0  23025  mdegmullem  23026  deg1leb  23043
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