MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dgrlb Unicode version

Theorem dgrlb 19618
Description: If all the coefficients above  M are zero, then the degree of  F is at most  M. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
dgrub.1  |-  A  =  (coeff `  F )
dgrub.2  |-  N  =  (deg `  F )
Assertion
Ref Expression
dgrlb  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )

Proof of Theorem dgrlb
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  NN0 )
2 dgrub.1 . . . . . . . . . . . . . 14  |-  A  =  (coeff `  F )
32dgrlem 19611 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
43simpld 445 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
543ad2ant1 976 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> ( S  u.  { 0 } ) )
6 ffn 5389 . . . . . . . . . . 11  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
7 elpreima 5645 . . . . . . . . . . 11  |-  ( A  Fn  NN0  ->  ( y  e.  ( `' A " ( CC  \  {
0 } ) )  <-> 
( y  e.  NN0  /\  ( A `  y
)  e.  ( CC 
\  { 0 } ) ) ) )
85, 6, 73syl 18 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( y  e.  ( `' A " ( CC 
\  { 0 } ) )  <->  ( y  e.  NN0  /\  ( A `
 y )  e.  ( CC  \  {
0 } ) ) ) )
98biimpa 470 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  (
y  e.  NN0  /\  ( A `  y )  e.  ( CC  \  { 0 } ) ) )
109simprd 449 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  ( A `  y )  e.  ( CC  \  {
0 } ) )
11 eldifsni 3750 . . . . . . . 8  |-  ( ( A `  y )  e.  ( CC  \  { 0 } )  ->  ( A `  y )  =/=  0
)
1210, 11syl 15 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  ( A `  y )  =/=  0 )
139simpld 445 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  NN0 )
14 simp3 957 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( A " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
152coef3 19614 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
16153ad2ant1 976 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> CC )
17 plyco0 19574 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. y  e.  NN0  ( ( A `
 y )  =/=  0  ->  y  <_  M ) ) )
181, 16, 17syl2anc 642 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( ( A "
( ZZ>= `  ( M  +  1 ) ) )  =  { 0 }  <->  A. y  e.  NN0  ( ( A `  y )  =/=  0  ->  y  <_  M )
) )
1914, 18mpbid 201 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  NN0  ( ( A `  y )  =/=  0  ->  y  <_  M )
)
2019r19.21bi 2641 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  NN0 )  -> 
( ( A `  y )  =/=  0  ->  y  <_  M )
)
2113, 20syldan 456 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  (
( A `  y
)  =/=  0  -> 
y  <_  M )
)
2212, 21mpd 14 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  <_  M )
2313nn0red 10019 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  RR )
241nn0red 10019 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  RR )
2524adantr 451 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  M  e.  RR )
2623, 25lenltd 8965 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  (
y  <_  M  <->  -.  M  <  y ) )
2722, 26mpbid 201 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  -.  M  <  y )
2827ralrimiva 2626 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  ( `' A " ( CC 
\  { 0 } ) )  -.  M  <  y )
29 nn0ssre 9969 . . . . . . 7  |-  NN0  C_  RR
30 ltso 8903 . . . . . . 7  |-  <  Or  RR
31 soss 4332 . . . . . . 7  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
3229, 30, 31mp2 17 . . . . . 6  |-  <  Or  NN0
3332a1i 10 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  <  Or  NN0 )
34 0z 10035 . . . . . . . 8  |-  0  e.  ZZ
3534a1i 10 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
36 cnvimass 5033 . . . . . . . 8  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
37 fdm 5393 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
384, 37syl 15 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
3936, 38syl5sseq 3226 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
403simprd 449 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
41 nn0uz 10262 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4241uzsupss 10310 . . . . . . 7  |-  ( ( 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 ) ) )
4335, 39, 40, 42syl3anc 1182 . . . . . 6  |-  ( 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 ) ) )
44433ad2ant1 976 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  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 ) ) )
4533, 44supnub 7213 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( ( M  e. 
NN0  /\  A. y  e.  ( `' A "
( CC  \  {
0 } ) )  -.  M  <  y
)  ->  -.  M  <  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) ) )
461, 28, 45mp2and 660 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  -.  M  <  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
47 dgrub.2 . . . . . 6  |-  N  =  (deg `  F )
482dgrval 19610 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
4947, 48syl5eq 2327 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
50493ad2ant1 976 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
5150breq2d 4035 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( M  <  N  <->  M  <  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) ) )
5246, 51mtbird 292 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  -.  M  <  N )
53 dgrcl 19615 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
5447, 53syl5eqel 2367 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
55543ad2ant1 976 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  NN0 )
5655nn0red 10019 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  RR )
5756, 24lenltd 8965 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( N  <_  M  <->  -.  M  <  N ) )
5852, 57mpbird 223 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )
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  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  coeidlem  19619  dgrle  19625  dgreq0  19646
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
  Copyright terms: Public domain W3C validator