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Theorem dgrub 20141
Description: If the  M-th coefficient of  F is nonzero, then the degree of  F is at least  M. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
dgrub.1  |-  A  =  (coeff `  F )
dgrub.2  |-  N  =  (deg `  F )
Assertion
Ref Expression
dgrub  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )

Proof of Theorem dgrub
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  F  e.  (Poly `  S )
)
2 simp2 958 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  NN0 )
3 dgrub.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
43coef3 20139 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
51, 4syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  A : NN0 --> CC )
65, 2ffvelrnd 5862 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  CC )
7 simp3 959 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  =/=  0 )
8 eldifsn 3919 . . . . . 6  |-  ( ( A `  M )  e.  ( CC  \  { 0 } )  <-> 
( ( A `  M )  e.  CC  /\  ( A `  M
)  =/=  0 ) )
96, 7, 8sylanbrc 646 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  ( CC  \  {
0 } ) )
103coef 20137 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
111, 10syl 16 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  A : NN0 --> ( S  u.  { 0 } ) )
12 ffn 5582 . . . . . 6  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
13 elpreima 5841 . . . . . 6  |-  ( A  Fn  NN0  ->  ( M  e.  ( `' A " ( CC  \  {
0 } ) )  <-> 
( M  e.  NN0  /\  ( A `  M
)  e.  ( CC 
\  { 0 } ) ) ) )
1411, 12, 133syl 19 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( M  e.  ( `' A " ( CC  \  { 0 } ) )  <->  ( M  e. 
NN0  /\  ( A `  M )  e.  ( CC  \  { 0 } ) ) ) )
152, 9, 14mpbir2and 889 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  ( `' A "
( CC  \  {
0 } ) ) )
16 nn0ssre 10214 . . . . . . 7  |-  NN0  C_  RR
17 ltso 9145 . . . . . . 7  |-  <  Or  RR
18 soss 4513 . . . . . . 7  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1916, 17, 18mp2 9 . . . . . 6  |-  <  Or  NN0
2019a1i 11 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  <  Or  NN0 )
21 0z 10282 . . . . . . 7  |-  0  e.  ZZ
2221a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
23 cnvimass 5215 . . . . . . 7  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
24 fdm 5586 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
2510, 24syl 16 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
2623, 25syl5sseq 3388 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
273dgrlem 20136 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
2827simprd 450 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
29 nn0uz 10509 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3029uzsupss 10557 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( `' A " ( CC 
\  { 0 } ) )  C_  NN0  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
)  ->  E. n  e.  NN0  ( A. x  e.  ( `' A "
( CC  \  {
0 } ) )  -.  n  <  x  /\  A. x  e.  NN0  ( x  <  n  ->  E. y  e.  ( `' A " ( CC 
\  { 0 } ) ) x  < 
y ) ) )
3122, 26, 28, 30syl3anc 1184 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  NN0  ( A. x  e.  ( `' A "
( CC  \  {
0 } ) )  -.  n  <  x  /\  A. x  e.  NN0  ( x  <  n  ->  E. y  e.  ( `' A " ( CC 
\  { 0 } ) ) x  < 
y ) ) )
3220, 31supub 7453 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( M  e.  ( `' A "
( CC  \  {
0 } ) )  ->  -.  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
331, 15, 32sylc 58 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M )
34 dgrub.2 . . . . . 6  |-  N  =  (deg `  F )
353dgrval 20135 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
3634, 35syl5eq 2479 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
371, 36syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  =  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
3837breq1d 4214 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( N  <  M  <->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
3933, 38mtbird 293 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  N  <  M )
402nn0red 10264 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  RR )
41 dgrcl 20140 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
4234, 41syl5eqel 2519 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
431, 42syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  NN0 )
4443nn0red 10264 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  RR )
4540, 44lenltd 9208 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
4639, 45mpbird 224 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    \ cdif 3309    u. cun 3310    C_ wss 3312   {csn 3806   class class class wbr 4204    Or wor 4494   `'ccnv 4868   dom cdm 4869   "cima 4872    Fn wfn 5440   -->wf 5441   ` cfv 5445   supcsup 7436   CCcc 8977   RRcr 8978   0cc0 8979    < clt 9109    <_ cle 9110   NN0cn0 10210   ZZcz 10271  Polycply 20091  coeffccoe 20093  degcdgr 20094
This theorem is referenced by:  dgrub2  20142  coeidlem  20144  coeid3  20147  dgreq  20151  coemullem  20156  coemulhi  20160  coemulc  20161  dgreq0  20171  dgrlt  20172  dgradd2  20174  dgrmul  20176  vieta1lem2  20216  aannenlem2  20234
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-fz 11033  df-fzo 11124  df-fl 11190  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-rlim 12271  df-sum 12468  df-0p 19550  df-ply 20095  df-coe 20097  df-dgr 20098
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