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Theorem mdegldg 19452
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 19449 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
873ad2ant2 977 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  =  sup ( ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )
)
92, 3mplrcl 16231 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1093ad2ant2 977 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  I  e.  _V )
115, 6tdeglem1 19444 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1210, 11syl 15 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H : A --> NN0 )
13 ffun 5391 . . . . . 6  |-  ( H : A --> NN0  ->  Fun 
H )
1412, 13syl 15 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Fun  H )
15 simp2 956 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  e.  B )
16 simp1 955 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Ring )
172, 3, 4, 15, 16mplelsfi 16232 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin )
18 imafi 7148 . . . . 5  |-  ( ( Fun  H  /\  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin )  ->  ( H "
( `' F "
( _V  \  {  .0.  } ) ) )  e.  Fin )
1914, 17, 18syl2anc 642 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) )  e.  Fin )
20 simp3 957 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  Y )
21 mdegldg.y . . . . . . . 8  |-  Y  =  ( 0g `  P
)
22 rnggrp 15346 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
23223ad2ant1 976 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Grp )
242, 5, 4, 21, 10, 23mpl0 16185 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Y  =  ( A  X.  {  .0.  } ) )
2520, 24neeqtrd 2468 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  ( A  X.  {  .0.  } ) )
26 eqid 2283 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
272, 26, 3, 5, 15mplelf 16178 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F : A --> ( Base `  R
) )
28 ffn 5389 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
2927, 28syl 15 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  Fn  A )
30 fvex 5539 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
314, 30eqeltri 2353 . . . . . . . 8  |-  .0.  e.  _V
32 fnsuppeq0 5733 . . . . . . . 8  |-  ( ( F  Fn  A  /\  .0.  e.  _V )  -> 
( ( `' F " ( _V  \  {  .0.  } ) )  =  (/) 
<->  F  =  ( A  X.  {  .0.  }
) ) )
3329, 31, 32sylancl 643 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( `' F "
( _V  \  {  .0.  } ) )  =  (/) 
<->  F  =  ( A  X.  {  .0.  }
) ) )
3433necon3bid 2481 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( `' F "
( _V  \  {  .0.  } ) )  =/=  (/) 
<->  F  =/=  ( A  X.  {  .0.  }
) ) )
3525, 34mpbird 223 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  =/=  (/) )
36 ffn 5389 . . . . . . . 8  |-  ( H : A --> NN0  ->  H  Fn  A )
3712, 36syl 15 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H  Fn  A )
38 cnvimass 5033 . . . . . . . 8  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
39 fdm 5393 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
4027, 39syl 15 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  dom  F  =  A )
4138, 40syl5sseq 3226 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)
42 fnimaeq0 5365 . . . . . . 7  |-  ( ( H  Fn  A  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  ( ( H " ( `' F " ( _V  \  {  .0.  } ) ) )  =  (/)  <->  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) ) )
4337, 41, 42syl2anc 642 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  =  (/) 
<->  ( `' F "
( _V  \  {  .0.  } ) )  =  (/) ) )
4443necon3bid 2481 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( `' F " ( _V 
\  {  .0.  }
) ) )  =/=  (/) 
<->  ( `' F "
( _V  \  {  .0.  } ) )  =/=  (/) ) )
4535, 44mpbird 223 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) )  =/=  (/) )
46 imassrn 5025 . . . . . 6  |-  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ran  H
47 frn 5395 . . . . . . 7  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
4812, 47syl 15 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ran  H 
C_  NN0 )
4946, 48syl5ss 3190 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  NN0 )
50 nn0ssre 9969 . . . . . 6  |-  NN0  C_  RR
51 ressxr 8876 . . . . . 6  |-  RR  C_  RR*
5250, 51sstri 3188 . . . . 5  |-  NN0  C_  RR*
5349, 52syl6ss 3191 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  RR* )
54 xrltso 10475 . . . . 5  |-  <  Or  RR*
55 fisupcl 7218 . . . . 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 651 . . . 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 1182 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  sup ( ( H "
( `' F "
( _V  \  {  .0.  } ) ) ) ,  RR* ,  <  )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
588, 57eqeltrd 2357 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
59 fvelimab 5578 . . . 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 642 . . 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 5650 . . . 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 15 . . 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 244 . 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 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    Or wor 4313    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   supcsup 7193   RRcr 8736   RR*cxr 8866    < clt 8867   NNcn 9746   NN0cn0 9965   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401   Grpcgrp 14362   Ringcrg 15337   mPoly cmpl 16089  ℂfldccnfld 16377   mDeg cmdg 19439
This theorem is referenced by:  mdegnn0cl  19457  deg1ldg  19478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-psr 16098  df-mpl 16100  df-cnfld 16378  df-mdeg 19441
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