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Theorem dgrub 22800
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 994 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  F  e.  (Poly `  S )
)
2 simp2 995 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  NN0 )
3 dgrub.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
43coef3 22798 . . . . . . . 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 6008 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  CC )
7 simp3 996 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  =/=  0 )
8 eldifsn 4141 . . . . . 6  |-  ( ( A `  M )  e.  ( CC  \  { 0 } )  <-> 
( ( A `  M )  e.  CC  /\  ( A `  M
)  =/=  0 ) )
96, 7, 8sylanbrc 662 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  ( CC  \  {
0 } ) )
103coef 22796 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
11 ffn 5713 . . . . . 6  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
12 elpreima 5983 . . . . . 6  |-  ( A  Fn  NN0  ->  ( M  e.  ( `' A " ( CC  \  {
0 } ) )  <-> 
( M  e.  NN0  /\  ( A `  M
)  e.  ( CC 
\  { 0 } ) ) ) )
131, 10, 11, 124syl 21 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( M  e.  ( `' A " ( CC  \  { 0 } ) )  <->  ( M  e. 
NN0  /\  ( A `  M )  e.  ( CC  \  { 0 } ) ) ) )
142, 9, 13mpbir2and 920 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  ( `' A "
( CC  \  {
0 } ) ) )
15 nn0ssre 10795 . . . . . . 7  |-  NN0  C_  RR
16 ltso 9654 . . . . . . 7  |-  <  Or  RR
17 soss 4807 . . . . . . 7  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1815, 16, 17mp2 9 . . . . . 6  |-  <  Or  NN0
1918a1i 11 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  <  Or  NN0 )
20 0zd 10872 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
21 cnvimass 5345 . . . . . . 7  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
22 fdm 5717 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
2310, 22syl 16 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
2421, 23syl5sseq 3537 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
253dgrlem 22795 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
2625simprd 461 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
27 nn0uz 11116 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2827uzsupss 11175 . . . . . 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 ) ) )
2920, 24, 26, 28syl3anc 1226 . . . . 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 ) ) )
3019, 29supub 7910 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( M  e.  ( `' A "
( CC  \  {
0 } ) )  ->  -.  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
311, 14, 30sylc 60 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M )
32 dgrub.2 . . . . . 6  |-  N  =  (deg `  F )
333dgrval 22794 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
3432, 33syl5eq 2507 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
351, 34syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  =  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
3635breq1d 4449 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( N  <  M  <->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
3731, 36mtbird 299 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  N  <  M )
382nn0red 10849 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  RR )
39 dgrcl 22799 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
4032, 39syl5eqel 2546 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
411, 40syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  NN0 )
4241nn0red 10849 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  RR )
4338, 42lenltd 9720 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
4437, 43mpbird 232 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    \ cdif 3458    u. cun 3459    C_ wss 3461   {csn 4016   class class class wbr 4439    Or wor 4788   `'ccnv 4987   dom cdm 4988   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570   supcsup 7892   CCcc 9479   RRcr 9480   0cc0 9481    < clt 9617    <_ cle 9618   NN0cn0 10791   ZZcz 10860  Polycply 22750  coeffccoe 22752  degcdgr 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-rlim 13397  df-sum 13594  df-0p 22246  df-ply 22754  df-coe 22756  df-dgr 22757
This theorem is referenced by:  dgrub2  22801  coeidlem  22803  coeid3  22806  dgreq  22810  coemullem  22816  coemulhi  22820  coemulc  22821  dgreq0  22831  dgrlt  22832  dgradd2  22834  dgrmul  22836  vieta1lem2  22876  aannenlem2  22894
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