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Theorem mdegleb 21540
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 21538 . . . 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 4307 . 2  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( ( D `  F )  <_  G  <->  sup ( ( H "
( F supp  .0.  )
) ,  RR* ,  <  )  <_  G ) )
10 imassrn 5185 . . . 4  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
112, 3mplrcl 17576 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1211adantr 465 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  I  e.  _V )
135, 6tdeglem1 21532 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1412, 13syl 16 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H : A --> NN0 )
15 frn 5570 . . . . . 6  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
1614, 15syl 16 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  NN0 )
17 nn0ssre 10588 . . . . . 6  |-  NN0  C_  RR
18 ressxr 9432 . . . . . 6  |-  RR  C_  RR*
1917, 18sstri 3370 . . . . 5  |-  NN0  C_  RR*
2016, 19syl6ss 3373 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  ran  H  C_  RR* )
2110, 20syl5ss 3372 . . 3  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( H " ( F supp  .0.  ) )  C_  RR* )
22 supxrleub 11294 . . 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 5564 . . . . 5  |-  ( H : A --> NN0  ->  H  Fn  A )
2514, 24syl 16 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  H  Fn  A )
26 suppssdm 6708 . . . . 5  |-  ( F supp 
.0.  )  C_  dom  F
27 eqid 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
28 simpl 457 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  e.  B )
292, 27, 3, 5, 28mplelf 17514 . . . . . 6  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F : A --> ( Base `  R ) )
30 fdm 5568 . . . . . 6  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
3129, 30syl 16 . . . . 5  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  dom  F  =  A )
3226, 31syl5sseq 3409 . . . 4  |-  ( ( F  e.  B  /\  G  e.  RR* )  -> 
( F supp  .0.  )  C_  A )
33 breq1 4300 . . . . 5  |-  ( y  =  ( H `  x )  ->  (
y  <_  G  <->  ( H `  x )  <_  G
) )
3433ralima 5962 . . . 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 5564 . . . . . . . 8  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3729, 36syl 16 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  F  Fn  A )
38 ovex 6121 . . . . . . . . . 10  |-  ( NN0 
^m  I )  e. 
_V
3938rabex 4448 . . . . . . . . 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 2527 . . . . . . 7  |-  ( ( F  e.  B  /\  G  e.  RR* )  ->  A  e.  _V )
42 fvex 5706 . . . . . . . . 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 6703 . . . . . . . 8  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
x  e.  ( F supp 
.0.  )  <->  ( x  e.  A  /\  ( F `  x )  =/=  .0.  ) ) )
46 fvex 5706 . . . . . . . . . . . 12  |-  ( F `
 x )  e. 
_V
4746biantrur 506 . . . . . . . . . . 11  |-  ( ( F `  x )  =/=  .0.  <->  ( ( F `  x )  e.  _V  /\  ( F `
 x )  =/= 
.0.  ) )
48 eldifsn 4005 . . . . . . . . . . 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 1218 . . . . . 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 754 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  G  e.  RR* )
5814ffvelrnda 5848 . . . . . . . . . . . 12  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  NN0 )
5919, 58sseldi 3359 . . . . . . . . . . 11  |-  ( ( ( F  e.  B  /\  G  e.  RR* )  /\  x  e.  A
)  ->  ( H `  x )  e.  RR* )
60 xrltnle 9448 . . . . . . . . . . 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 2617 . . . . . . . . . . . 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 2742 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   {crab 2724   _Vcvv 2977    \ cdif 3330    C_ wss 3333   {csn 3882   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   supp csupp 6695    ^m cmap 7219   Fincfn 7315   supcsup 7695   RRcr 9286   RR*cxr 9422    < clt 9423    <_ cle 9424   NNcn 10327   NN0cn0 10584   Basecbs 14179   0gc0g 14383    gsumg cgsu 14384   mPoly cmpl 17425  ℂfldccnfld 17823   mDeg cmdg 21527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-0g 14385  df-gsum 14386  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-cntz 15840  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-cring 16653  df-psr 17428  df-mpl 17430  df-cnfld 17824  df-mdeg 21529
This theorem is referenced by:  mdeglt  21541  mdegaddle  21550  mdegvscale  21551  mdegle0  21553  mdegmullem  21554  deg1leb  21571
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