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Theorem dgrub 19616
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 955 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  F  e.  (Poly `  S )
)
2 simp2 956 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  NN0 )
3 dgrub.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
43coef3 19614 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
51, 4syl 15 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  A : NN0 --> CC )
6 ffvelrn 5663 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  M  e.  NN0 )  -> 
( A `  M
)  e.  CC )
75, 2, 6syl2anc 642 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  CC )
8 simp3 957 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  =/=  0 )
9 eldifsn 3749 . . . . . 6  |-  ( ( A `  M )  e.  ( CC  \  { 0 } )  <-> 
( ( A `  M )  e.  CC  /\  ( A `  M
)  =/=  0 ) )
107, 8, 9sylanbrc 645 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  ( CC  \  {
0 } ) )
113coef 19612 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
121, 11syl 15 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  A : NN0 --> ( S  u.  { 0 } ) )
13 ffn 5389 . . . . . 6  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
14 elpreima 5645 . . . . . 6  |-  ( A  Fn  NN0  ->  ( M  e.  ( `' A " ( CC  \  {
0 } ) )  <-> 
( M  e.  NN0  /\  ( A `  M
)  e.  ( CC 
\  { 0 } ) ) ) )
1512, 13, 143syl 18 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( M  e.  ( `' A " ( CC  \  { 0 } ) )  <->  ( M  e. 
NN0  /\  ( A `  M )  e.  ( CC  \  { 0 } ) ) ) )
162, 10, 15mpbir2and 888 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  ( `' A "
( CC  \  {
0 } ) ) )
17 nn0ssre 9969 . . . . . . 7  |-  NN0  C_  RR
18 ltso 8903 . . . . . . 7  |-  <  Or  RR
19 soss 4332 . . . . . . 7  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
2017, 18, 19mp2 17 . . . . . 6  |-  <  Or  NN0
2120a1i 10 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  <  Or  NN0 )
22 0z 10035 . . . . . . 7  |-  0  e.  ZZ
2322a1i 10 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
24 cnvimass 5033 . . . . . . 7  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
25 fdm 5393 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
2611, 25syl 15 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
2724, 26syl5sseq 3226 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
283dgrlem 19611 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
2928simprd 449 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
30 nn0uz 10262 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3130uzsupss 10310 . . . . . 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 ) ) )
3223, 27, 29, 31syl3anc 1182 . . . . 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 ) ) )
3321, 32supub 7210 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( M  e.  ( `' A "
( CC  \  {
0 } ) )  ->  -.  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
341, 16, 33sylc 56 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M )
35 dgrub.2 . . . . . 6  |-  N  =  (deg `  F )
363dgrval 19610 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
3735, 36syl5eq 2327 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
381, 37syl 15 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  =  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
3938breq1d 4033 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( N  <  M  <->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
4034, 39mtbird 292 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  N  <  M )
412nn0red 10019 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  RR )
42 dgrcl 19615 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
4335, 42syl5eqel 2367 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
441, 43syl 15 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  NN0 )
4544nn0red 10019 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  RR )
4641, 45lenltd 8965 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
4740, 46mpbird 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640   class class class wbr 4023    Or wor 4313   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868   NN0cn0 9965   ZZcz 10024  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dgrub2  19617  coeidlem  19619  coeid3  19622  dgreq  19626  coemullem  19631  coemulhi  19635  coemulc  19636  dgreq0  19646  dgrlt  19647  dgradd2  19649  dgrmul  19651  vieta1lem2  19691  aannenlem2  19709
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-inf2 7342  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-pre-sup 8815  ax-addf 8816
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-pm 6775  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-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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