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Theorem mdegldg 22196
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 22192 . . . 4  |-  ( F  e.  B  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
873ad2ant2 1013 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  =  sup ( ( H
" ( F supp  .0.  ) ) ,  RR* ,  <  ) )
92, 3mplrcl 17920 . . . . . . . 8  |-  ( F  e.  B  ->  I  e.  _V )
1093ad2ant2 1013 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  I  e.  _V )
115, 6tdeglem1 22186 . . . . . . 7  |-  ( I  e.  _V  ->  H : A --> NN0 )
1210, 11syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H : A --> NN0 )
13 ffun 5726 . . . . . 6  |-  ( H : A --> NN0  ->  Fun 
H )
1412, 13syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Fun  H )
15 simp2 992 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  e.  B )
16 simp1 991 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Ring )
172, 3, 4, 15, 16mplelsfi 17921 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F finSupp  .0.  )
18 fsuppimp 7826 . . . . . . 7  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
1918simprd 463 . . . . . 6  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
2017, 19syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( F supp  .0.  )  e.  Fin )
21 imafi 7804 . . . . 5  |-  ( ( Fun  H  /\  ( F supp  .0.  )  e.  Fin )  ->  ( H "
( F supp  .0.  )
)  e.  Fin )
2214, 20, 21syl2anc 661 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  e.  Fin )
23 simp3 993 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  Y )
24 mdegldg.y . . . . . . . 8  |-  Y  =  ( 0g `  P
)
25 rnggrp 16986 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
26253ad2ant1 1012 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  R  e.  Grp )
272, 5, 4, 24, 10, 26mpl0 17869 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  Y  =  ( A  X.  {  .0.  } ) )
2823, 27neeqtrd 2757 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  =/=  ( A  X.  {  .0.  } ) )
29 eqid 2462 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
302, 29, 3, 5, 15mplelf 17858 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F : A --> ( Base `  R
) )
31 ffn 5724 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  F  Fn  A )
3230, 31syl 16 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  F  Fn  A )
33 fvex 5869 . . . . . . . . 9  |-  ( 0g
`  R )  e. 
_V
344, 33eqeltri 2546 . . . . . . . 8  |-  .0.  e.  _V
35 ovex 6302 . . . . . . . . . . 11  |-  ( NN0 
^m  I )  e. 
_V
3635rabex 4593 . . . . . . . . . 10  |-  { m  e.  ( NN0  ^m  I
)  |  ( `' m " NN )  e.  Fin }  e.  _V
375, 36eqeltri 2546 . . . . . . . . 9  |-  A  e. 
_V
38 fnsuppeq0 6920 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  A  e.  _V  /\  .0.  e.  _V )  ->  (
( F supp  .0.  )  =  (/)  <->  F  =  ( A  X.  {  .0.  }
) ) )
3937, 38mp3an2 1307 . . . . . . . 8  |-  ( ( F  Fn  A  /\  .0.  e.  _V )  -> 
( ( F supp  .0.  )  =  (/)  <->  F  =  ( A  X.  {  .0.  } ) ) )
4032, 34, 39sylancl 662 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( F supp  .0.  )  =  (/)  <->  F  =  ( A  X.  {  .0.  }
) ) )
4140necon3bid 2720 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( F supp  .0.  )  =/=  (/)  <->  F  =/=  ( A  X.  {  .0.  }
) ) )
4228, 41mpbird 232 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( F supp  .0.  )  =/=  (/) )
43 ffn 5724 . . . . . . . 8  |-  ( H : A --> NN0  ->  H  Fn  A )
4412, 43syl 16 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  H  Fn  A )
45 suppssdm 6906 . . . . . . . 8  |-  ( F supp 
.0.  )  C_  dom  F
46 fdm 5728 . . . . . . . . 9  |-  ( F : A --> ( Base `  R )  ->  dom  F  =  A )
4730, 46syl 16 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  dom  F  =  A )
4845, 47syl5sseq 3547 . . . . . . 7  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( F supp  .0.  )  C_  A
)
49 fnimaeq0 5695 . . . . . . 7  |-  ( ( H  Fn  A  /\  ( F supp  .0.  )  C_  A )  ->  (
( H " ( F supp  .0.  ) )  =  (/) 
<->  ( F supp  .0.  )  =  (/) ) )
5044, 48, 49syl2anc 661 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( F supp  .0.  ) )  =  (/) 
<->  ( F supp  .0.  )  =  (/) ) )
5150necon3bid 2720 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  (
( H " ( F supp  .0.  ) )  =/=  (/) 
<->  ( F supp  .0.  )  =/=  (/) ) )
5242, 51mpbird 232 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  =/=  (/) )
53 imassrn 5341 . . . . . 6  |-  ( H
" ( F supp  .0.  ) )  C_  ran  H
54 frn 5730 . . . . . . 7  |-  ( H : A --> NN0  ->  ran 
H  C_  NN0 )
5512, 54syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ran  H 
C_  NN0 )
5653, 55syl5ss 3510 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  C_  NN0 )
57 nn0ssre 10790 . . . . . 6  |-  NN0  C_  RR
58 ressxr 9628 . . . . . 6  |-  RR  C_  RR*
5957, 58sstri 3508 . . . . 5  |-  NN0  C_  RR*
6056, 59syl6ss 3511 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( H " ( F supp  .0.  ) )  C_  RR* )
61 xrltso 11338 . . . . 5  |-  <  Or  RR*
62 fisupcl 7918 . . . . 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.  ) ) )
6361, 62mpan 670 . . . 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.  ) ) )
6422, 52, 60, 63syl3anc 1223 . . 3  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  sup ( ( H "
( F supp  .0.  )
) ,  RR* ,  <  )  e.  ( H "
( F supp  .0.  )
) )
658, 64eqeltrd 2550 . 2  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
Y )  ->  ( D `  F )  e.  ( H " ( F supp  .0.  ) ) )
66 fvelimab 5916 . . . 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 ) ) )
6744, 48, 66syl2anc 661 . . 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 ) ) )
68 rexsupp 6910 . . . . 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 ) ) ) )
6937, 34, 68mp3an23 1311 . . . 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 ) ) ) )
7032, 69syl 16 . . 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 ) ) ) )
7167, 70bitrd 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
) ) ) )
7265, 71mpbid 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810   {crab 2813   _Vcvv 3108    C_ wss 3471   (/)c0 3780   {csn 4022   class class class wbr 4442    |-> cmpt 4500    Or wor 4794    X. cxp 4992   `'ccnv 4993   dom cdm 4994   ran crn 4995   "cima 4997   Fun wfun 5575    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   supp csupp 6893    ^m cmap 7412   Fincfn 7508   finSupp cfsupp 7820   supcsup 7891   RRcr 9482   RR*cxr 9618    < clt 9619   NNcn 10527   NN0cn0 10786   Basecbs 14481   0gc0g 14686    gsumg cgsu 14687   Grpcgrp 15718   Ringcrg 16981   mPoly cmpl 17768  ℂfldccnfld 18186   mDeg cmdg 22181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-fzo 11784  df-seq 12066  df-hash 12363  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-0g 14688  df-gsum 14689  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-subg 15988  df-cntz 16145  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-cring 16984  df-psr 17771  df-mpl 17773  df-cnfld 18187  df-mdeg 22183
This theorem is referenced by:  mdegnn0cl  22201  deg1ldg  22222
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