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Theorem dgrlb 22923
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 998 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  NN0 )
2 dgrub.1 . . . . . . . . . . . . . 14  |-  A  =  (coeff `  F )
32dgrlem 22916 . . . . . . . . . . . . 13  |-  ( F  e.  (Poly `  S
)  ->  ( A : NN0 --> ( S  u.  { 0 } )  /\  E. n  e.  ZZ  A. x  e.  ( `' A " ( CC  \  { 0 } ) ) x  <_  n
) )
43simpld 457 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> ( S  u.  {
0 } ) )
543ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> ( S  u.  { 0 } ) )
6 ffn 5713 . . . . . . . . . . 11  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
7 elpreima 5984 . . . . . . . . . . 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 482 . . . . . . . . 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 461 . . . . . . . 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 4097 . . . . . . . 8  |-  ( ( A `  y )  e.  ( CC  \  { 0 } )  ->  ( A `  y )  =/=  0
)
1210, 11syl 17 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  ( A `  y )  =/=  0 )
139simpld 457 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  NN0 )
14 simp3 999 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( A " ( ZZ>=
`  ( M  + 
1 ) ) )  =  { 0 } )
152coef3 22919 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
16153ad2ant1 1018 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A : NN0 --> CC )
17 plyco0 22879 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 }  <->  A. y  e.  NN0  ( ( A `
 y )  =/=  0  ->  y  <_  M ) ) )
181, 16, 17syl2anc 659 . . . . . . . . . 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 2772 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  NN0 )  -> 
( ( A `  y )  =/=  0  ->  y  <_  M )
)
2113, 20syldan 468 . . . . . . 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 10893 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  y  e.  RR )
241nn0red 10893 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  M  e.  RR )
2524adantr 463 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  /\  y  e.  ( `' A " ( CC  \  { 0 } ) ) )  ->  M  e.  RR )
2623, 25lenltd 9762 . . . . . 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 2817 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  A. y  e.  ( `' A " ( CC 
\  { 0 } ) )  -.  M  <  y )
29 nn0ssre 10839 . . . . . . 7  |-  NN0  C_  RR
30 ltso 9695 . . . . . . 7  |-  <  Or  RR
31 soss 4761 . . . . . . 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 10916 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  0  e.  ZZ )
35 cnvimass 5176 . . . . . . . 8  |-  ( `' A " ( CC 
\  { 0 } ) )  C_  dom  A
36 fdm 5717 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
374, 36syl 17 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  dom  A  = 
NN0 )
3835, 37syl5sseq 3489 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( `' A " ( CC  \  { 0 } ) )  C_  NN0 )
393simprd 461 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  ZZ  A. x  e.  ( `' A "
( CC  \  {
0 } ) ) x  <_  n )
40 nn0uz 11160 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
4140uzsupss 11218 . . . . . . 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 1230 . . . . . 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 1018 . . . . 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 7954 . . . 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 677 . . 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 22915 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
4846, 47syl5eq 2455 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
49483ad2ant1 1018 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
5049breq2d 4406 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  -> 
( M  <  N  <->  M  <  sup ( ( `' A " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) ) )
5145, 50mtbird 299 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  -.  M  <  N )
52 dgrcl 22920 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
5346, 52syl5eqel 2494 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
54533ad2ant1 1018 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  NN0 )
5554nn0red 10893 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A
" ( ZZ>= `  ( M  +  1 ) ) )  =  {
0 } )  ->  N  e.  RR )
5655, 24lenltd 9762 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754    \ cdif 3410    u. cun 3411    C_ wss 3413   {csn 3971   class class class wbr 4394    Or wor 4742   `'ccnv 4821   dom cdm 4822   "cima 4825    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277   supcsup 7933   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    < clt 9657    <_ cle 9658   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126  Polycply 22871  coeffccoe 22873  degcdgr 22874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-rlim 13459  df-sum 13656  df-0p 22367  df-ply 22875  df-coe 22877  df-dgr 22878
This theorem is referenced by:  coeidlem  22924  dgrle  22930  dgreq0  22952
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