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Theorem dgrlb 19634
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 19627 . . . . . . . . . . . . 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 5405 . . . . . . . . . . 11  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
7 elpreima 5661 . . . . . . . . . . 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 3763 . . . . . . . 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 19630 . . . . . . . . . . . 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 19590 . . . . . . . . . . 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 2654 . . . . . . . 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 10035 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  RR )
241nn0red 10035 . . . . . . . 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 8981 . . . . . 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 2639 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  ( `' A " ( CC 
\  { 0 } ) )  -.  M  <  y )
29 nn0ssre 9985 . . . . . . 7  |-  NN0  C_  RR
30 ltso 8919 . . . . . . 7  |-  <  Or  RR
31 soss 4348 . . . . . . 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 10051 . . . . . . . 8  |-  0  e.  ZZ
3534a1i 10 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
36 cnvimass 5049 . . . . . . . 8  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
37 fdm 5409 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
384, 37syl 15 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
3936, 38syl5sseq 3239 . . . . . . 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 10278 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4241uzsupss 10326 . . . . . . 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 7229 . . . 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 19626 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
4947, 48syl5eq 2340 . . . . 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 4051 . . 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 19631 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
5447, 53syl5eqel 2380 . . . . 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 10035 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  RR )
5756, 24lenltd 8981 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653   class class class wbr 4039    Or wor 4329   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  coeidlem  19635  dgrle  19641  dgreq0  19662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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