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Theorem dgrub 22361
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 991 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  F  e.  (Poly `  S )
)
2 simp2 992 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  NN0 )
3 dgrub.1 . . . . . . . . 9  |-  A  =  (coeff `  F )
43coef3 22359 . . . . . . . 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 6015 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  CC )
7 simp3 993 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  =/=  0 )
8 eldifsn 4147 . . . . . 6  |-  ( ( A `  M )  e.  ( CC  \  { 0 } )  <-> 
( ( A `  M )  e.  CC  /\  ( A `  M
)  =/=  0 ) )
96, 7, 8sylanbrc 664 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( A `  M )  e.  ( CC  \  {
0 } ) )
103coef 22357 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
11 ffn 5724 . . . . . 6  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
12 elpreima 5994 . . . . . 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 915 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  ( `' A "
( CC  \  {
0 } ) ) )
15 nn0ssre 10790 . . . . . . 7  |-  NN0  C_  RR
16 ltso 9656 . . . . . . 7  |-  <  Or  RR
17 soss 4813 . . . . . . 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 10867 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
21 cnvimass 5350 . . . . . . 7  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
22 fdm 5728 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
2310, 22syl 16 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
2421, 23syl5sseq 3547 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
253dgrlem 22356 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
2625simprd 463 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
27 nn0uz 11107 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
2827uzsupss 11165 . . . . . 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 1223 . . . . 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 22355 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
3432, 33syl5eq 2515 . . . . 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 4452 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  ( N  <  M  <->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  <  M ) )
3731, 36mtbird 301 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  -.  N  <  M )
382nn0red 10844 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  e.  RR )
39 dgrcl 22360 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
4032, 39syl5eqel 2554 . . . . 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 10844 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  N  e.  RR )
4338, 42lenltd 9721 . 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810    \ cdif 3468    u. cun 3469    C_ wss 3471   {csn 4022   class class class wbr 4442    Or wor 4794   `'ccnv 4993   dom cdm 4994   "cima 4997    Fn wfn 5576   -->wf 5577   ` cfv 5581   supcsup 7891   CCcc 9481   RRcr 9482   0cc0 9483    < clt 9619    <_ cle 9620   NN0cn0 10786   ZZcz 10855  Polycply 22311  coeffccoe 22313  degcdgr 22314
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-inf2 8049  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-pre-sup 9561  ax-addf 9562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  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-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  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-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-sum 13460  df-0p 21807  df-ply 22315  df-coe 22317  df-dgr 22318
This theorem is referenced by:  dgrub2  22362  coeidlem  22364  coeid3  22367  dgreq  22371  coemullem  22376  coemulhi  22380  coemulc  22381  dgreq0  22391  dgrlt  22392  dgradd2  22394  dgrmul  22396  vieta1lem2  22436  aannenlem2  22454
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