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Theorem plyeq0 19593
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 19572 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 8831 . . . . . . . . . 10  |-  0  e.  CC
43a1i 10 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
54snssd 3760 . . . . . . . 8  |-  ( ph  ->  { 0 }  C_  CC )
62, 5unssd 3351 . . . . . . 7  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
7 cnex 8818 . . . . . . 7  |-  CC  e.  _V
8 ssexg 4160 . . . . . . 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 9971 . . . . . 6  |-  NN0  e.  _V
11 elmapg 6785 . . . . . 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 5389 . . . 4  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
1513, 14syl 15 . . 3  |-  ( ph  ->  A  Fn  NN0 )
16 imadmrn 5024 . . . 4  |-  ( A
" dom  A )  =  ran  A
17 fdm 5393 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
18 fimacnv 5657 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  ( `' A " ( S  u.  { 0 } ) )  =  NN0 )
1917, 18eqtr4d 2318 . . . . . . . 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 2283 . . . . . . . . . . . 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 19592 . . . . . . . . . . 11  |-  -.  ( ph  /\  ( `' A " ( S  \  {
0 } ) )  =/=  (/) )
3332pm2.21i 123 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =  (/) )
3421, 33pm2.61dane 2524 . . . . . . . . 9  |-  ( ph  ->  ( `' A "
( S  \  {
0 } ) )  =  (/) )
3534uneq1d 3328 . . . . . . . 8  |-  ( ph  ->  ( ( `' A " ( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )  =  ( (/)  u.  ( `' A " { 0 } ) ) )
36 undif1 3529 . . . . . . . . . 10  |-  ( ( S  \  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
3736imaeq2i 5010 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( `' A " ( S  u.  {
0 } ) )
38 imaundi 5093 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( ( `' A " ( S 
\  { 0 } ) )  u.  ( `' A " { 0 } ) )
3937, 38eqtr3i 2305 . . . . . . . 8  |-  ( `' A " ( S  u.  { 0 } ) )  =  ( ( `' A "
( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )
40 un0 3479 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( `' A " { 0 } )
41 uncom 3319 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( (/)  u.  ( `' A " { 0 } ) )
4240, 41eqtr3i 2305 . . . . . . . 8  |-  ( `' A " { 0 } )  =  (
(/)  u.  ( `' A " { 0 } ) )
4335, 39, 423eqtr4g 2340 . . . . . . 7  |-  ( ph  ->  ( `' A "
( S  u.  {
0 } ) )  =  ( `' A " { 0 } ) )
4420, 43eqtrd 2315 . . . . . 6  |-  ( ph  ->  dom  A  =  ( `' A " { 0 } ) )
45 eqimss 3230 . . . . . 6  |-  ( dom 
A  =  ( `' A " { 0 } )  ->  dom  A 
C_  ( `' A " { 0 } ) )
4644, 45syl 15 . . . . 5  |-  ( ph  ->  dom  A  C_  ( `' A " { 0 } ) )
47 ffun 5391 . . . . . . 7  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  Fun  A )
4813, 47syl 15 . . . . . 6  |-  ( ph  ->  Fun  A )
49 ssid 3197 . . . . . 6  |-  dom  A  C_ 
dom  A
50 funimass3 5641 . . . . . 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 3223 . . 3  |-  ( ph  ->  ran  A  C_  { 0 } )
54 df-f 5259 . . 3  |-  ( A : NN0 --> { 0 }  <->  ( A  Fn  NN0 
/\  ran  A  C_  { 0 } ) )
5515, 53, 54sylanbrc 645 . 2  |-  ( ph  ->  A : NN0 --> { 0 } )
56 c0ex 8832 . . 3  |-  0  e.  _V
5756fconst2 5730 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867   NN0cn0 9965   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158   0 pc0p 19024
This theorem is referenced by:  coeeulem  19606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025
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