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Theorem dgrub 20153
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 20151 . . . . . . . 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 5871 . . . . . 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 3927 . . . . . 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 20149 . . . . . . 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 5591 . . . . . 6  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
13 elpreima 5850 . . . . . 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 10225 . . . . . . 7  |-  NN0  C_  RR
17 ltso 9156 . . . . . . 7  |-  <  Or  RR
18 soss 4521 . . . . . . 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 10293 . . . . . . 7  |-  0  e.  ZZ
2221a1i 11 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
23 cnvimass 5224 . . . . . . 7  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
24 fdm 5595 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
2510, 24syl 16 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
2623, 25syl5sseq 3396 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
273dgrlem 20148 . . . . . . 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 10520 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
3029uzsupss 10568 . . . . . 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 7464 . . . 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 20147 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
3634, 35syl5eq 2480 . . . . 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 4222 . . 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 10275 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  RR )
41 dgrcl 20152 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
4234, 41syl5eqel 2520 . . . . 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 10275 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  RR )
4540, 44lenltd 9219 . 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 2599   A.wral 2705   E.wrex 2706    \ cdif 3317    u. cun 3318    C_ wss 3320   {csn 3814   class class class wbr 4212    Or wor 4502   `'ccnv 4877   dom cdm 4878   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990    < clt 9120    <_ cle 9121   NN0cn0 10221   ZZcz 10282  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  dgrub2  20154  coeidlem  20156  coeid3  20159  dgreq  20163  coemullem  20168  coemulhi  20172  coemulc  20173  dgreq0  20183  dgrlt  20184  dgradd2  20186  dgrmul  20188  vieta1lem2  20228  aannenlem2  20246
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110
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