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Theorem dgrlb 20155
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 958 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  NN0 )
2 dgrub.1 . . . . . . . . . . . . . 14  |-  A  =  (coeff `  F )
32dgrlem 20148 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
43simpld 446 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
543ad2ant1 978 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> ( S  u.  { 0 } ) )
6 ffn 5591 . . . . . . . . . . 11  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
7 elpreima 5850 . . . . . . . . . . 11  |-  ( A  Fn  NN0  ->  ( y  e.  ( `' A " ( CC  \  {
0 } ) )  <-> 
( y  e.  NN0  /\  ( A `  y
)  e.  ( CC 
\  { 0 } ) ) ) )
85, 6, 73syl 19 . . . . . . . . . 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 471 . . . . . . . . 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 450 . . . . . . . 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 3928 . . . . . . . 8  |-  ( ( A `  y )  e.  ( CC  \  { 0 } )  ->  ( A `  y )  =/=  0
)
1210, 11syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  ( A `  y )  =/=  0 )
139simpld 446 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  NN0 )
14 simp3 959 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( A " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
152coef3 20151 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
16153ad2ant1 978 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> CC )
17 plyco0 20111 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. y  e.  NN0  ( ( A `
 y )  =/=  0  ->  y  <_  M ) ) )
181, 16, 17syl2anc 643 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  NN0  ( ( A `  y )  =/=  0  ->  y  <_  M )
)
2019r19.21bi 2804 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  NN0 )  -> 
( ( A `  y )  =/=  0  ->  y  <_  M )
)
2113, 20syldan 457 . . . . . . 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 15 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  <_  M )
2313nn0red 10275 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  RR )
241nn0red 10275 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  RR )
2524adantr 452 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  M  e.  RR )
2623, 25lenltd 9219 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  (
y  <_  M  <->  -.  M  <  y ) )
2722, 26mpbid 202 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  -.  M  <  y )
2827ralrimiva 2789 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  ( `' A " ( CC 
\  { 0 } ) )  -.  M  <  y )
29 nn0ssre 10225 . . . . . . 7  |-  NN0  C_  RR
30 ltso 9156 . . . . . . 7  |-  <  Or  RR
31 soss 4521 . . . . . . 7  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
3229, 30, 31mp2 9 . . . . . 6  |-  <  Or  NN0
3332a1i 11 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  <  Or  NN0 )
34 0z 10293 . . . . . . . 8  |-  0  e.  ZZ
3534a1i 11 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
36 cnvimass 5224 . . . . . . . 8  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
37 fdm 5595 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
384, 37syl 16 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
3936, 38syl5sseq 3396 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
403simprd 450 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
41 nn0uz 10520 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4241uzsupss 10568 . . . . . . 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 1184 . . . . . 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 978 . . . . 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 7467 . . . 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 661 . . 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 20147 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
4947, 48syl5eq 2480 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
50493ad2ant1 978 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
5150breq2d 4224 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( M  <  N  <->  M  <  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) ) )
5246, 51mtbird 293 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  -.  M  <  N )
53 dgrcl 20152 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
5447, 53syl5eqel 2520 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
55543ad2ant1 978 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  NN0 )
5655nn0red 10275 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  RR )
5756, 24lenltd 9219 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( N  <_  M  <->  -.  M  <  N ) )
5852, 57mpbird 224 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 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  (class class class)co 6081   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  coeidlem  20156  dgrle  20162  dgreq0  20183
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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