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Theorem mdegldg 22756
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 22752 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
873ad2ant2 1019 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
92, 3mplrcl 18472 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1093ad2ant2 1019 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  I  e.  _V )
115, 6tdeglem1 22746 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1210, 11syl 17 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H : A --> NN0 )
13 ffun 5715 . . . . . 6  |-  ( H : A --> NN0  ->  Fun 
H )
1412, 13syl 17 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Fun  H )
15 simp2 998 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  e.  B )
16 simp1 997 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Ring )
172, 3, 4, 15, 16mplelsfi 18473 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F finSupp  .0.  )
1817fsuppimpd 7869 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( F supp  .0.  )  e.  Fin )
19 imafi 7846 . . . . 5  |-  ( ( Fun  H  /\  ( F supp  .0.  )  e.  Fin )  ->  ( H "
( F supp  .0.  )
)  e.  Fin )
2014, 18, 19syl2anc 659 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  e.  Fin )
21 simp3 999 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  Y )
22 mdegldg.y . . . . . . . 8  |-  Y  =  ( 0g `  P
)
23 ringgrp 17521 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
24233ad2ant1 1018 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Grp )
252, 5, 4, 22, 10, 24mpl0 18421 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Y  =  ( A  X.  {  .0.  } ) )
2621, 25neeqtrd 2698 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  ( A  X.  {  .0.  } ) )
27 eqid 2402 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
282, 27, 3, 5, 15mplelf 18410 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F : A --> ( Base `  R
) )
29 ffn 5713 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3028, 29syl 17 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  Fn  A )
31 fvex 5858 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
324, 31eqeltri 2486 . . . . . . . 8  |-  .0.  e.  _V
33 ovex 6305 . . . . . . . . . 10  |-  ( NN0 
^m  I )  e. 
_V
345, 33rabex2 4546 . . . . . . . . 9  |-  A  e. 
_V
35 fnsuppeq0 6930 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
( F supp  .0.  )  =  (/)  <->  F  =  ( A  X.  {  .0.  }
) ) )
3634, 35mp3an2 1314 . . . . . . . 8  |-  ( ( F  Fn  A  /\  .0.  e.  _V )  -> 
( ( F supp  .0.  )  =  (/)  <->  F  =  ( A  X.  {  .0.  } ) ) )
3730, 32, 36sylancl 660 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( F supp  .0.  )  =  (/)  <->  F  =  ( A  X.  {  .0.  }
) ) )
3837necon3bid 2661 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( F supp  .0.  )  =/=  (/)  <->  F  =/=  ( A  X.  {  .0.  }
) ) )
3926, 38mpbird 232 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( F supp  .0.  )  =/=  (/) )
40 ffn 5713 . . . . . . . 8  |-  ( H : A --> NN0  ->  H  Fn  A )
4112, 40syl 17 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H  Fn  A )
42 suppssdm 6914 . . . . . . . 8  |-  ( F supp 
.0.  )  C_  dom  F
43 fdm 5717 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
4428, 43syl 17 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  dom  F  =  A )
4542, 44syl5sseq 3489 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( F supp  .0.  )  C_  A
)
46 fnimaeq0 5682 . . . . . . 7  |-  ( ( H  Fn  A  /\  ( F supp  .0.  )  C_  A )  ->  (
( H " ( F supp  .0.  ) )  =  (/) 
<->  ( F supp  .0.  )  =  (/) ) )
4741, 45, 46syl2anc 659 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( F supp  .0.  ) )  =  (/) 
<->  ( F supp  .0.  )  =  (/) ) )
4847necon3bid 2661 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( F supp  .0.  ) )  =/=  (/) 
<->  ( F supp  .0.  )  =/=  (/) ) )
4939, 48mpbird 232 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  =/=  (/) )
50 imassrn 5167 . . . . . 6  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
51 frn 5719 . . . . . . 7  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
5212, 51syl 17 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ran  H 
C_  NN0 )
5350, 52syl5ss 3452 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  C_  NN0 )
54 nn0ssre 10839 . . . . . 6  |-  NN0  C_  RR
55 ressxr 9666 . . . . . 6  |-  RR  C_  RR*
5654, 55sstri 3450 . . . . 5  |-  NN0  C_  RR*
5753, 56syl6ss 3453 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  C_  RR* )
58 xrltso 11399 . . . . 5  |-  <  Or  RR*
59 fisupcl 7960 . . . . 5  |-  ( (  <  Or  RR*  /\  (
( H " ( F supp  .0.  ) )  e. 
Fin  /\  ( H " ( F supp  .0.  )
)  =/=  (/)  /\  ( H " ( F supp  .0.  ) )  C_  RR* )
)  ->  sup (
( H " ( F supp  .0.  ) ) , 
RR* ,  <  )  e.  ( H " ( F supp  .0.  ) ) )
6058, 59mpan 668 . . . 4  |-  ( ( ( H " ( F supp  .0.  ) )  e. 
Fin  /\  ( H " ( F supp  .0.  )
)  =/=  (/)  /\  ( H " ( F supp  .0.  ) )  C_  RR* )  ->  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  )  e.  ( H " ( F supp 
.0.  ) ) )
6120, 49, 57, 60syl3anc 1230 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  sup ( ( H "
( F supp  .0.  )
) ,  RR* ,  <  )  e.  ( H "
( F supp  .0.  )
) )
628, 61eqeltrd 2490 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  e.  ( H " ( F supp  .0.  ) ) )
63 fvelimab 5904 . . . 4  |-  ( ( H  Fn  A  /\  ( F supp  .0.  )  C_  A )  ->  (
( D `  F
)  e.  ( H
" ( F supp  .0.  ) )  <->  E. x  e.  ( F supp  .0.  )
( H `  x
)  =  ( D `
 F ) ) )
6441, 45, 63syl2anc 659 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( D `  F
)  e.  ( H
" ( F supp  .0.  ) )  <->  E. x  e.  ( F supp  .0.  )
( H `  x
)  =  ( D `
 F ) ) )
65 rexsupp 6920 . . . . 5  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  ( E. x  e.  ( F supp  .0.  ) ( H `
 x )  =  ( D `  F
)  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x
)  =  ( D `
 F ) ) ) )
6634, 32, 65mp3an23 1318 . . . 4  |-  ( F  Fn  A  ->  ( E. x  e.  ( F supp  .0.  ) ( H `
 x )  =  ( D `  F
)  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x
)  =  ( D `
 F ) ) ) )
6730, 66syl 17 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( E. x  e.  ( F supp  .0.  ) ( H `
 x )  =  ( D `  F
)  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `  x
)  =  ( D `
 F ) ) ) )
6864, 67bitrd 253 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( D `  F
)  e.  ( H
" ( F supp  .0.  ) )  <->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `
 x )  =  ( D `  F
) ) ) )
6962, 68mpbid 210 1  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  E. x  e.  A  ( ( F `  x )  =/=  .0.  /\  ( H `
 x )  =  ( D `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   {crab 2757   _Vcvv 3058    C_ wss 3413   (/)c0 3737   {csn 3971    |-> cmpt 4452    Or wor 4742    X. cxp 4820   `'ccnv 4821   dom cdm 4822   ran crn 4823   "cima 4825   Fun wfun 5562    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   supp csupp 6901    ^m cmap 7456   Fincfn 7553   supcsup 7933   RRcr 9520   RR*cxr 9656    < clt 9657   NNcn 10575   NN0cn0 10835   Basecbs 14839   0gc0g 15052    gsumg cgsu 15053   Grpcgrp 16375   Ringcrg 17516   mPoly cmpl 18320  ℂfldccnfld 18738   mDeg cmdg 22741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-fz 11725  df-fzo 11853  df-seq 12150  df-hash 12451  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-0g 15054  df-gsum 15055  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-grp 16379  df-minusg 16380  df-subg 16520  df-cntz 16677  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-psr 18323  df-mpl 18325  df-cnfld 18739  df-mdeg 22743
This theorem is referenced by:  mdegnn0cl  22761  deg1ldg  22782
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