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Theorem plyeq0 21678
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 21657 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  ->  0p  =  ( 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 0cnd 9378 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
43snssd 4017 . . . . . . . 8  |-  ( ph  ->  { 0 }  C_  CC )
52, 4unssd 3531 . . . . . . 7  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
6 cnex 9362 . . . . . . 7  |-  CC  e.  _V
7 ssexg 4437 . . . . . . 7  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
85, 6, 7sylancl 662 . . . . . 6  |-  ( ph  ->  ( S  u.  {
0 } )  e. 
_V )
9 nn0ex 10584 . . . . . 6  |-  NN0  e.  _V
10 elmapg 7226 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( A  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
118, 9, 10sylancl 662 . . . . 5  |-  ( ph  ->  ( A  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  A : NN0 --> ( S  u.  { 0 } ) ) )
121, 11mpbid 210 . . . 4  |-  ( ph  ->  A : NN0 --> ( S  u.  { 0 } ) )
13 ffn 5558 . . . 4  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
1412, 13syl 16 . . 3  |-  ( ph  ->  A  Fn  NN0 )
15 imadmrn 5178 . . . 4  |-  ( A
" dom  A )  =  ran  A
16 fdm 5562 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
17 fimacnv 5834 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  ( `' A " ( S  u.  { 0 } ) )  =  NN0 )
1816, 17eqtr4d 2477 . . . . . . . 8  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  =  ( `' A "
( S  u.  {
0 } ) ) )
1912, 18syl 16 . . . . . . 7  |-  ( ph  ->  dom  A  =  ( `' A " ( S  u.  { 0 } ) ) )
20 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =  (/) )
212adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  ->  S  C_  CC )
22 plyeq0.2 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN0 )
2322adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  ->  N  e.  NN0 )
241adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  ->  A  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
25 plyeq0.4 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
2625adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
27 plyeq0.5 . . . . . . . . . . . . 13  |-  ( ph  ->  0p  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
2827adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
0p  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( z ^ k ) ) ) )
29 eqid 2442 . . . . . . . . . . . 12  |-  sup (
( `' A "
( S  \  {
0 } ) ) ,  RR ,  <  )  =  sup ( ( `' A " ( S 
\  { 0 } ) ) ,  RR ,  <  )
30 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =/=  (/) )
3121, 23, 24, 26, 28, 29, 30plyeq0lem 21677 . . . . . . . . . . 11  |-  -.  ( ph  /\  ( `' A " ( S  \  {
0 } ) )  =/=  (/) )
3231pm2.21i 131 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =  (/) )
3320, 32pm2.61dane 2688 . . . . . . . . 9  |-  ( ph  ->  ( `' A "
( S  \  {
0 } ) )  =  (/) )
3433uneq1d 3508 . . . . . . . 8  |-  ( ph  ->  ( ( `' A " ( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )  =  ( (/)  u.  ( `' A " { 0 } ) ) )
35 undif1 3753 . . . . . . . . . 10  |-  ( ( S  \  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
3635imaeq2i 5166 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( `' A " ( S  u.  {
0 } ) )
37 imaundi 5248 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( ( `' A " ( S 
\  { 0 } ) )  u.  ( `' A " { 0 } ) )
3836, 37eqtr3i 2464 . . . . . . . 8  |-  ( `' A " ( S  u.  { 0 } ) )  =  ( ( `' A "
( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )
39 un0 3661 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( `' A " { 0 } )
40 uncom 3499 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( (/)  u.  ( `' A " { 0 } ) )
4139, 40eqtr3i 2464 . . . . . . . 8  |-  ( `' A " { 0 } )  =  (
(/)  u.  ( `' A " { 0 } ) )
4234, 38, 413eqtr4g 2499 . . . . . . 7  |-  ( ph  ->  ( `' A "
( S  u.  {
0 } ) )  =  ( `' A " { 0 } ) )
4319, 42eqtrd 2474 . . . . . 6  |-  ( ph  ->  dom  A  =  ( `' A " { 0 } ) )
44 eqimss 3407 . . . . . 6  |-  ( dom 
A  =  ( `' A " { 0 } )  ->  dom  A 
C_  ( `' A " { 0 } ) )
4543, 44syl 16 . . . . 5  |-  ( ph  ->  dom  A  C_  ( `' A " { 0 } ) )
46 ffun 5560 . . . . . . 7  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  Fun  A )
4712, 46syl 16 . . . . . 6  |-  ( ph  ->  Fun  A )
48 ssid 3374 . . . . . 6  |-  dom  A  C_ 
dom  A
49 funimass3 5818 . . . . . 6  |-  ( ( Fun  A  /\  dom  A 
C_  dom  A )  ->  ( ( A " dom  A )  C_  { 0 }  <->  dom  A  C_  ( `' A " { 0 } ) ) )
5047, 48, 49sylancl 662 . . . . 5  |-  ( ph  ->  ( ( A " dom  A )  C_  { 0 }  <->  dom  A  C_  ( `' A " { 0 } ) ) )
5145, 50mpbird 232 . . . 4  |-  ( ph  ->  ( A " dom  A )  C_  { 0 } )
5215, 51syl5eqssr 3400 . . 3  |-  ( ph  ->  ran  A  C_  { 0 } )
53 df-f 5421 . . 3  |-  ( A : NN0 --> { 0 }  <->  ( A  Fn  NN0 
/\  ran  A  C_  { 0 } ) )
5414, 52, 53sylanbrc 664 . 2  |-  ( ph  ->  A : NN0 --> { 0 } )
55 c0ex 9379 . . 3  |-  0  e.  _V
5655fconst2 5933 . 2  |-  ( A : NN0 --> { 0 }  <->  A  =  ( NN0  X.  { 0 } ) )
5754, 56sylib 196 1  |-  ( ph  ->  A  =  ( NN0 
X.  { 0 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   _Vcvv 2971    \ cdif 3324    u. cun 3325    C_ wss 3327   (/)c0 3636   {csn 3876    e. cmpt 4349    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842   Fun wfun 5411    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090    ^m cmap 7213   supcsup 7689   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286    < clt 9417   NN0cn0 10578   ZZ>=cuz 10860   ...cfz 11436   ^cexp 11864   sum_csu 13162   0pc0p 21146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-0p 21147
This theorem is referenced by:  coeeulem  21691
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