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Theorem dgrlb 22368
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 997 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  NN0 )
2 dgrub.1 . . . . . . . . . . . . . 14  |-  A  =  (coeff `  F )
32dgrlem 22361 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
43simpld 459 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
543ad2ant1 1017 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> ( S  u.  { 0 } ) )
6 ffn 5729 . . . . . . . . . . 11  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
7 elpreima 5999 . . . . . . . . . . 11  |-  ( A  Fn  NN0  ->  ( y  e.  ( `' A " ( CC  \  {
0 } ) )  <-> 
( y  e.  NN0  /\  ( A `  y
)  e.  ( CC 
\  { 0 } ) ) ) )
85, 6, 73syl 20 . . . . . . . . . 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 484 . . . . . . . . 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 463 . . . . . . . 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 4153 . . . . . . . 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 459 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  NN0 )
14 simp3 998 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( A " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
152coef3 22364 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
16153ad2ant1 1017 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> CC )
17 plyco0 22324 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. y  e.  NN0  ( ( A `
 y )  =/=  0  ->  y  <_  M ) ) )
181, 16, 17syl2anc 661 . . . . . . . . . 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 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  NN0  ( ( A `  y )  =/=  0  ->  y  <_  M )
)
2019r19.21bi 2833 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  NN0 )  -> 
( ( A `  y )  =/=  0  ->  y  <_  M )
)
2113, 20syldan 470 . . . . . . 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 10849 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  RR )
241nn0red 10849 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  RR )
2524adantr 465 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  M  e.  RR )
2623, 25lenltd 9726 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  (
y  <_  M  <->  -.  M  <  y ) )
2722, 26mpbid 210 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  -.  M  <  y )
2827ralrimiva 2878 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  ( `' A " ( CC 
\  { 0 } ) )  -.  M  <  y )
29 nn0ssre 10795 . . . . . . 7  |-  NN0  C_  RR
30 ltso 9661 . . . . . . 7  |-  <  Or  RR
31 soss 4818 . . . . . . 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 0zd 10872 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
35 cnvimass 5355 . . . . . . . 8  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
36 fdm 5733 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
374, 36syl 16 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
3835, 37syl5sseq 3552 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
393simprd 463 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
40 nn0uz 11112 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4140uzsupss 11170 . . . . . . 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 ) ) )
4234, 38, 39, 41syl3anc 1228 . . . . . 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 ) ) )
43423ad2ant1 1017 . . . . 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 ) ) )
4433, 43supnub 7918 . . . 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 ,  <  ) ) )
451, 28, 44mp2and 679 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  -.  M  <  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
46 dgrub.2 . . . . . 6  |-  N  =  (deg `  F )
472dgrval 22360 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
4846, 47syl5eq 2520 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
49483ad2ant1 1017 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
5049breq2d 4459 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( M  <  N  <->  M  <  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) ) )
5145, 50mtbird 301 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  -.  M  <  N )
52 dgrcl 22365 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
5346, 52syl5eqel 2559 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
54533ad2ant1 1017 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  NN0 )
5554nn0red 10849 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  RR )
5655, 24lenltd 9726 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( N  <_  M  <->  -.  M  <  N ) )
5751, 56mpbird 232 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  <_  M )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    \ cdif 3473    u. cun 3474    C_ wss 3476   {csn 4027   class class class wbr 4447    Or wor 4799   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    <_ cle 9625   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078  Polycply 22316  coeffccoe 22318  degcdgr 22319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-0p 21812  df-ply 22320  df-coe 22322  df-dgr 22323
This theorem is referenced by:  coeidlem  22369  dgrle  22375  dgreq0  22396
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