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Theorem mdegldg 19505
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 19502 . . . 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 16280 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1093ad2ant2 977 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  I  e.  _V )
115, 6tdeglem1 19497 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1210, 11syl 15 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H : A --> NN0 )
13 ffun 5429 . . . . . 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 16281 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin )
18 imafi 7193 . . . . 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 15395 . . . . . . . . 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 16234 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Y  =  ( A  X.  {  .0.  } ) )
2520, 24neeqtrd 2501 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  ( A  X.  {  .0.  } ) )
26 eqid 2316 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
272, 26, 3, 5, 15mplelf 16227 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F : A --> ( Base `  R
) )
28 ffn 5427 . . . . . . . . 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 5577 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
314, 30eqeltri 2386 . . . . . . . 8  |-  .0.  e.  _V
32 fnsuppeq0 5774 . . . . . . . 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 2514 . . . . . 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 5427 . . . . . . . 8  |-  ( H : A --> NN0  ->  H  Fn  A )
3712, 36syl 15 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H  Fn  A )
38 cnvimass 5070 . . . . . . . 8  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
39 fdm 5431 . . . . . . . . 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 3260 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)
42 fnimaeq0 5402 . . . . . . 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 2514 . . . . 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 5062 . . . . . 6  |-  ( H
" ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ran  H
47 frn 5433 . . . . . . 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 3224 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  NN0 )
50 nn0ssre 10016 . . . . . 6  |-  NN0  C_  RR
51 ressxr 8921 . . . . . 6  |-  RR  C_  RR*
5250, 51sstri 3222 . . . . 5  |-  NN0  C_  RR*
5349, 52syl6ss 3225 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  RR* )
54 xrltso 10522 . . . . 5  |-  <  Or  RR*
55 fisupcl 7263 . . . . 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 2390 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  e.  ( H " ( `' F " ( _V 
\  {  .0.  }
) ) ) )
59 fvelimab 5616 . . . 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 5688 . . . 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 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   {crab 2581   _Vcvv 2822    \ cdif 3183    C_ wss 3186   (/)c0 3489   {csn 3674    e. cmpt 4114    Or wor 4350    X. cxp 4724   `'ccnv 4725   dom cdm 4726   ran crn 4727   "cima 4729   Fun wfun 5286    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    ^m cmap 6815   Fincfn 6906   supcsup 7238   RRcr 8781   RR*cxr 8911    < clt 8912   NNcn 9791   NN0cn0 10012   Basecbs 13195   0gc0g 13449    gsumg cgsu 13450   Grpcgrp 14411   Ringcrg 15386   mPoly cmpl 16138  ℂfldccnfld 16432   mDeg cmdg 19492
This theorem is referenced by:  mdegnn0cl  19510  deg1ldg  19531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-fzo 10918  df-seq 11094  df-hash 11385  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-0g 13453  df-gsum 13454  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-subg 14667  df-cntz 14842  df-cmn 15140  df-abl 15141  df-mgp 15375  df-rng 15389  df-cring 15390  df-ur 15391  df-psr 16147  df-mpl 16149  df-cnfld 16433  df-mdeg 19494
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