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

Theorem mdegldg 19989
Description: A nonzero polynomial has some coefficient which witnesses its degree. (Contributed by Stefan O'Rear, 23-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 ) )
mdegldg.y  |-  Y  =  ( 0g `  P
)
Assertion
Ref Expression
mdegldg  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `
 x )  =  ( D `  F
) ) )
Distinct variable groups:    A, h    m, I    .0. , h    x, A   
x, B    x, F    x, H    h, I    x, R    x,  .0.    h, m    x, D
Allowed substitution hints:    A( m)    B( h, m)    D( h, m)    P( x, h, m)    R( h, m)    F( h, m)    H( h, m)    I( x)    Y( x, h, m)    .0. ( m)

Proof of Theorem mdegldg
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 19986 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
873ad2ant2 979 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
92, 3mplrcl 16550 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1093ad2ant2 979 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  I  e.  _V )
115, 6tdeglem1 19981 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1210, 11syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H : A --> NN0 )
13 ffun 5593 . . . . . 6  |-  ( H : A --> NN0  ->  Fun 
H )
1412, 13syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Fun  H )
15 simp2 958 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  e.  B )
16 simp1 957 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Ring )
172, 3, 4, 15, 16mplelsfi 16551 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin )
18 imafi 7399 . . . . 5  |-  ( ( Fun  H  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin )  ->  ( H "
( `' F "
( _V  \  {  .0.  } ) ) )  e.  Fin )
1914, 17, 18syl2anc 643 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) )  e.  Fin )
20 simp3 959 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  Y )
21 mdegldg.y . . . . . . . 8  |-  Y  =  ( 0g `  P
)
22 rnggrp 15669 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
23223ad2ant1 978 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Grp )
242, 5, 4, 21, 10, 23mpl0 16504 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Y  =  ( A  X.  {  .0.  } ) )
2520, 24neeqtrd 2623 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  ( A  X.  {  .0.  } ) )
26 eqid 2436 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
272, 26, 3, 5, 15mplelf 16497 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F : A --> ( Base `  R
) )
28 ffn 5591 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
2927, 28syl 16 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  Fn  A )
30 fvex 5742 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
314, 30eqeltri 2506 . . . . . . . 8  |-  .0.  e.  _V
32 fnsuppeq0 5953 . . . . . . . 8  |-  ( ( F  Fn  A  /\  .0.  e.  _V )  -> 
( ( `' F " ( _V  \  {  .0.  } ) )  =  (/) 
<->  F  =  ( A  X.  {  .0.  }
) ) )
3329, 31, 32sylancl 644 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( `' F "
( _V  \  {  .0.  } ) )  =  (/) 
<->  F  =  ( A  X.  {  .0.  }
) ) )
3433necon3bid 2636 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( `' F "
( _V  \  {  .0.  } ) )  =/=  (/) 
<->  F  =/=  ( A  X.  {  .0.  }
) ) )
3525, 34mpbird 224 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  =/=  (/) )
36 ffn 5591 . . . . . . . 8  |-  ( H : A --> NN0  ->  H  Fn  A )
3712, 36syl 16 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H  Fn  A )
38 cnvimass 5224 . . . . . . . 8  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
39 fdm 5595 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
4027, 39syl 16 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  dom  F  =  A )
4138, 40syl5sseq 3396 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)
42 fnimaeq0 5566 . . . . . . 7  |-  ( ( H  Fn  A  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  ( ( H " ( `' F " ( _V  \  {  .0.  } ) ) )  =  (/)  <->  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) ) )
4337, 41, 42syl2anc 643 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  =  (/) 
<->  ( `' F "
( _V  \  {  .0.  } ) )  =  (/) ) )
4443necon3bid 2636 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  =/=  (/) 
<->  ( `' F "
( _V  \  {  .0.  } ) )  =/=  (/) ) )
4535, 44mpbird 224 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) )  =/=  (/) )
46 imassrn 5216 . . . . . 6  |-  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ran  H
47 frn 5597 . . . . . . 7  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
4812, 47syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ran  H 
C_  NN0 )
4946, 48syl5ss 3359 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  NN0 )
50 nn0ssre 10225 . . . . . 6  |-  NN0  C_  RR
51 ressxr 9129 . . . . . 6  |-  RR  C_  RR*
5250, 51sstri 3357 . . . . 5  |-  NN0  C_  RR*
5349, 52syl6ss 3360 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  RR* )
54 xrltso 10734 . . . . 5  |-  <  Or  RR*
55 fisupcl 7472 . . . . 5  |-  ( (  <  Or  RR*  /\  (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  e. 
Fin  /\  ( H " ( `' F "
( _V  \  {  .0.  } ) ) )  =/=  (/)  /\  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  RR* ) )  ->  sup ( ( H "
( `' F "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
5654, 55mpan 652 . . . 4  |-  ( ( ( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  e. 
Fin  /\  ( H " ( `' F "
( _V  \  {  .0.  } ) ) )  =/=  (/)  /\  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  RR* )  ->  sup ( ( H "
( `' F "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
5719, 45, 53, 56syl3anc 1184 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  sup ( ( H "
( `' F "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
588, 57eqeltrd 2510 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
59 fvelimab 5782 . . . 4  |-  ( ( H  Fn  A  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  ( ( D `  F )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  <->  E. x  e.  ( `' F "
( _V  \  {  .0.  } ) ) ( H `  x )  =  ( D `  F ) ) )
6037, 41, 59syl2anc 643 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( D `  F
)  e.  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) )  <->  E. x  e.  ( `' F " ( _V 
\  {  .0.  }
) ) ( H `
 x )  =  ( D `  F
) ) )
61 rexsupp 5855 . . . 4  |-  ( F  Fn  A  ->  ( E. x  e.  ( `' F " ( _V 
\  {  .0.  }
) ) ( H `
 x )  =  ( D `  F
)  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x
)  =  ( D `
 F ) ) ) )
6229, 61syl 16 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( E. x  e.  ( `' F " ( _V 
\  {  .0.  }
) ) ( H `
 x )  =  ( D `  F
)  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x
)  =  ( D `
 F ) ) ) )
6360, 62bitrd 245 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( D `  F
)  e.  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) )  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x
)  =  ( D `
 F ) ) ) )
6458, 63mpbid 202 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `
 x )  =  ( D `  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   {csn 3814    e. cmpt 4266    Or wor 4502    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   supcsup 7445   RRcr 8989   RR*cxr 9119    < clt 9120   NNcn 10000   NN0cn0 10221   Basecbs 13469   0gc0g 13723    gsumg cgsu 13724   Grpcgrp 14685   Ringcrg 15660   mPoly cmpl 16408  ℂfldccnfld 16703   mDeg cmdg 19976
This theorem is referenced by:  mdegnn0cl  19994  deg1ldg  20015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-gsum 13728  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-subg 14941  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-psr 16417  df-mpl 16419  df-cnfld 16704  df-mdeg 19978
  Copyright terms: Public domain W3C validator