MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plyeq0 Unicode version

Theorem plyeq0 19808
Description: If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 19787 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
plyeq0.1  |-  ( ph  ->  S  C_  CC )
plyeq0.2  |-  ( ph  ->  N  e.  NN0 )
plyeq0.3  |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
plyeq0.4  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
plyeq0.5  |-  ( ph  ->  0 p  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
Assertion
Ref Expression
plyeq0  |-  ( ph  ->  A  =  ( NN0 
X.  { 0 } ) )
Distinct variable groups:    z, k, A    k, N, z    ph, k,
z    S, k, z

Proof of Theorem plyeq0
StepHypRef Expression
1 plyeq0.3 . . . . 5  |-  ( ph  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
2 plyeq0.1 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
3 0cn 8978 . . . . . . . . . 10  |-  0  e.  CC
43a1i 10 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
54snssd 3858 . . . . . . . 8  |-  ( ph  ->  { 0 }  C_  CC )
62, 5unssd 3439 . . . . . . 7  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
7 cnex 8965 . . . . . . 7  |-  CC  e.  _V
8 ssexg 4262 . . . . . . 7  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
96, 7, 8sylancl 643 . . . . . 6  |-  ( ph  ->  ( S  u.  {
0 } )  e. 
_V )
10 nn0ex 10120 . . . . . 6  |-  NN0  e.  _V
11 elmapg 6928 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
129, 10, 11sylancl 643 . . . . 5  |-  ( ph  ->  ( A  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
131, 12mpbid 201 . . . 4  |-  ( ph  ->  A : NN0 --> ( S  u.  { 0 } ) )
14 ffn 5495 . . . 4  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
1513, 14syl 15 . . 3  |-  ( ph  ->  A  Fn  NN0 )
16 imadmrn 5127 . . . 4  |-  ( A
" dom  A )  =  ran  A
17 fdm 5499 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
18 fimacnv 5764 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  ( `' A " ( S  u.  { 0 } ) )  =  NN0 )
1917, 18eqtr4d 2401 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  =  ( `' A "
( S  u.  {
0 } ) ) )
2013, 19syl 15 . . . . . . 7  |-  ( ph  ->  dom  A  =  ( `' A " ( S  u.  { 0 } ) ) )
21 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =  (/) )
222adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  ->  S  C_  CC )
23 plyeq0.2 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN0 )
2423adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  ->  N  e.  NN0 )
251adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
26 plyeq0.4 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
2726adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
28 plyeq0.5 . . . . . . . . . . . . 13  |-  ( ph  ->  0 p  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
2928adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
0 p  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
30 eqid 2366 . . . . . . . . . . . 12  |-  sup (
( `' A "
( S  \  {
0 } ) ) ,  RR ,  <  )  =  sup ( ( `' A " ( S 
\  { 0 } ) ) ,  RR ,  <  )
31 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =/=  (/) )
3222, 24, 25, 27, 29, 30, 31plyeq0lem 19807 . . . . . . . . . . 11  |-  -.  ( ph  /\  ( `' A " ( S  \  {
0 } ) )  =/=  (/) )
3332pm2.21i 123 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =  (/) )
3421, 33pm2.61dane 2607 . . . . . . . . 9  |-  ( ph  ->  ( `' A "
( S  \  {
0 } ) )  =  (/) )
3534uneq1d 3416 . . . . . . . 8  |-  ( ph  ->  ( ( `' A " ( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )  =  ( (/)  u.  ( `' A " { 0 } ) ) )
36 undif1 3618 . . . . . . . . . 10  |-  ( ( S  \  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
3736imaeq2i 5113 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( `' A " ( S  u.  {
0 } ) )
38 imaundi 5196 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( ( `' A " ( S 
\  { 0 } ) )  u.  ( `' A " { 0 } ) )
3937, 38eqtr3i 2388 . . . . . . . 8  |-  ( `' A " ( S  u.  { 0 } ) )  =  ( ( `' A "
( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )
40 un0 3567 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( `' A " { 0 } )
41 uncom 3407 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( (/)  u.  ( `' A " { 0 } ) )
4240, 41eqtr3i 2388 . . . . . . . 8  |-  ( `' A " { 0 } )  =  (
(/)  u.  ( `' A " { 0 } ) )
4335, 39, 423eqtr4g 2423 . . . . . . 7  |-  ( ph  ->  ( `' A "
( S  u.  {
0 } ) )  =  ( `' A " { 0 } ) )
4420, 43eqtrd 2398 . . . . . 6  |-  ( ph  ->  dom  A  =  ( `' A " { 0 } ) )
45 eqimss 3316 . . . . . 6  |-  ( dom 
A  =  ( `' A " { 0 } )  ->  dom  A 
C_  ( `' A " { 0 } ) )
4644, 45syl 15 . . . . 5  |-  ( ph  ->  dom  A  C_  ( `' A " { 0 } ) )
47 ffun 5497 . . . . . . 7  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  Fun  A )
4813, 47syl 15 . . . . . 6  |-  ( ph  ->  Fun  A )
49 ssid 3283 . . . . . 6  |-  dom  A  C_ 
dom  A
50 funimass3 5748 . . . . . 6  |-  ( ( Fun  A  /\  dom  A 
C_  dom  A )  ->  ( ( A " dom  A )  C_  { 0 }  <->  dom  A  C_  ( `' A " { 0 } ) ) )
5148, 49, 50sylancl 643 . . . . 5  |-  ( ph  ->  ( ( A " dom  A )  C_  { 0 }  <->  dom  A  C_  ( `' A " { 0 } ) ) )
5246, 51mpbird 223 . . . 4  |-  ( ph  ->  ( A " dom  A )  C_  { 0 } )
5316, 52syl5eqssr 3309 . . 3  |-  ( ph  ->  ran  A  C_  { 0 } )
54 df-f 5362 . . 3  |-  ( A : NN0 --> { 0 }  <->  ( A  Fn  NN0 
/\  ran  A  C_  { 0 } ) )
5515, 53, 54sylanbrc 645 . 2  |-  ( ph  ->  A : NN0 --> { 0 } )
56 c0ex 8979 . . 3  |-  0  e.  _V
5756fconst2 5848 . 2  |-  ( A : NN0 --> { 0 }  <->  A  =  ( NN0  X.  { 0 } ) )
5855, 57sylib 188 1  |-  ( ph  ->  A  =  ( NN0 
X.  { 0 } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   _Vcvv 2873    \ cdif 3235    u. cun 3236    C_ wss 3238   (/)c0 3543   {csn 3729    e. cmpt 4179    X. cxp 4790   `'ccnv 4791   dom cdm 4792   ran crn 4793   "cima 4795   Fun wfun 5352    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981    ^m cmap 6915   supcsup 7340   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014   NN0cn0 10114   ZZ>=cuz 10381   ...cfz 10935   ^cexp 11269   sum_csu 12366   0 pc0p 19239
This theorem is referenced by:  coeeulem  19821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-0p 19240
  Copyright terms: Public domain W3C validator