MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mdegleb Structured version   Unicode version

Theorem mdegleb 22589
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 22587 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
87adantr 465 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( D `  F
)  =  sup (
( H " ( F supp  .0.  ) ) , 
RR* ,  <  ) )
98breq1d 4466 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  sup ( ( H "
( F supp  .0.  )
) ,  RR* ,  <  )  <_  G ) )
10 imassrn 5358 . . . 4  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
112, 3mplrcl 18280 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1211adantr 465 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  I  e.  _V )
135, 6tdeglem1 22581 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1412, 13syl 16 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H : A --> NN0 )
15 frn 5743 . . . . . 6  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
1614, 15syl 16 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  NN0 )
17 nn0ssre 10820 . . . . . 6  |-  NN0  C_  RR
18 ressxr 9654 . . . . . 6  |-  RR  C_  RR*
1917, 18sstri 3508 . . . . 5  |-  NN0  C_  RR*
2016, 19syl6ss 3511 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  RR* )
2110, 20syl5ss 3510 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( H " ( F supp  .0.  ) )  C_  RR* )
22 supxrleub 11543 . . 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 667 . 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 5737 . . . . 5  |-  ( H : A --> NN0  ->  H  Fn  A )
2514, 24syl 16 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H  Fn  A )
26 suppssdm 6930 . . . . 5  |-  ( F supp 
.0.  )  C_  dom  F
27 eqid 2457 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
28 simpl 457 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  e.  B )
292, 27, 3, 5, 28mplelf 18218 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F : A --> ( Base `  R ) )
30 fdm 5741 . . . . . 6  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
3129, 30syl 16 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  dom  F  =  A )
3226, 31syl5sseq 3547 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( F supp  .0.  )  C_  A )
33 breq1 4459 . . . . 5  |-  ( y  =  ( H `  x )  ->  (
y  <_  G  <->  ( H `  x )  <_  G
) )
3433ralima 6153 . . . 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 661 . . 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 5737 . . . . . . . 8  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3729, 36syl 16 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  Fn  A )
38 ovex 6324 . . . . . . . . . 10  |-  ( NN0 
^m  I )  e. 
_V
3938rabex 4607 . . . . . . . . 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 2549 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  A  e.  _V )
42 fvex 5882 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
434, 42eqeltri 2541 . . . . . . . 8  |-  .0.  e.  _V
4443a1i 11 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  .0.  e.  _V )
45 elsuppfn 6925 . . . . . . . 8  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
x  e.  ( F supp 
.0.  )  <->  ( x  e.  A  /\  ( F `  x )  =/=  .0.  ) ) )
46 fvex 5882 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4746biantrur 506 . . . . . . . . . . 11  |-  ( ( F `  x )  =/=  .0.  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
48 eldifsn 4157 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
4947, 48bitr4i 252 . . . . . . . . . 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 703 . . . . . . . 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 253 . . . . . . 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 1228 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( x  e.  ( F supp  .0.  )  <->  ( x  e.  A  /\  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) ) )
5453imbi1d 317 . . . . 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 446 . . . . . 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 292 . . . . . . . 8  |-  ( ( ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  ->  ( H `  x )  <_  G
)  <->  ( -.  ( H `  x )  <_  G  ->  -.  ( F `  x )  e.  ( _V  \  {  .0.  } ) ) )
57 simplr 755 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  G  e.  RR* )
5814ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  NN0 )
5919, 58sseldi 3497 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  RR* )
60 xrltnle 9670 . . . . . . . . . . 11  |-  ( ( G  e.  RR*  /\  ( H `  x )  e.  RR* )  ->  ( G  <  ( H `  x )  <->  -.  ( H `  x )  <_  G ) )
6157, 59, 60syl2anc 661 . . . . . . . . . 10  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( G  <  ( H `  x
)  <->  -.  ( H `  x )  <_  G
) )
6261bicomd 201 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( H `  x )  <_  G  <->  G  <  ( H `  x ) ) )
63 ianor 488 . . . . . . . . . . 11  |-  ( -.  ( ( F `  x )  e.  _V  /\  ( F `  x
)  =/=  .0.  )  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
6463, 48xchnxbir 309 . . . . . . . . . 10  |-  ( -.  ( F `  x
)  e.  ( _V 
\  {  .0.  }
)  <->  ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) )
65 orcom 387 . . . . . . . . . . . 12  |-  ( ( -.  ( F `  x )  e.  _V  \/  -.  ( F `  x )  =/=  .0.  ) 
<->  ( -.  ( F `
 x )  =/= 
.0.  \/  -.  ( F `  x )  e.  _V ) )
6646notnoti 123 . . . . . . . . . . . . 13  |-  -.  -.  ( F `  x )  e.  _V
6766biorfi 407 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( -.  ( F `  x )  =/=  .0.  \/  -.  ( F `  x )  e.  _V ) )
68 nne 2658 . . . . . . . . . . . 12  |-  ( -.  ( F `  x
)  =/=  .0.  <->  ( F `  x )  =  .0.  )
6965, 67, 683bitr2i 273 . . . . . . . . . . 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 257 . . . . . . . . 9  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( -.  ( F `  x )  e.  ( _V  \  {  .0.  } )  <->  ( F `  x )  =  .0.  ) )
7262, 71imbi12d 320 . . . . . . . 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 257 . . . . . . 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 687 . . . . . 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 257 . . . . 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 253 . . . 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 2892 . . 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 253 . 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 279 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468    C_ wss 3471   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supp csupp 6917    ^m cmap 7438   Fincfn 7535   supcsup 7918   RRcr 9508   RR*cxr 9644    < clt 9645    <_ cle 9646   NNcn 10556   NN0cn0 10816   Basecbs 14643   0gc0g 14856    gsumg cgsu 14857   mPoly cmpl 18128  ℂfldccnfld 18546   mDeg cmdg 22576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-0g 14858  df-gsum 14859  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-cntz 16481  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-cring 17327  df-psr 18131  df-mpl 18133  df-cnfld 18547  df-mdeg 22578
This theorem is referenced by:  mdeglt  22590  mdegaddle  22599  mdegvscale  22600  mdegle0  22602  mdegmullem  22603  deg1leb  22620
  Copyright terms: Public domain W3C validator