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Theorem plyeq0 22734
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 22713 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 9606 . . . . . . . . 9  |-  ( ph  ->  0  e.  CC )
43snssd 4177 . . . . . . . 8  |-  ( ph  ->  { 0 }  C_  CC )
52, 4unssd 3676 . . . . . . 7  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
6 cnex 9590 . . . . . . 7  |-  CC  e.  _V
7 ssexg 4602 . . . . . . 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 10822 . . . . . 6  |-  NN0  e.  _V
10 elmapg 7451 . . . . . 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 5737 . . . 4  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  A  Fn  NN0 )
1412, 13syl 16 . . 3  |-  ( ph  ->  A  Fn  NN0 )
15 imadmrn 5357 . . . 4  |-  ( A
" dom  A )  =  ran  A
16 fdm 5741 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  dom  A  = 
NN0 )
17 fimacnv 6020 . . . . . . . . 9  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  ( `' A " ( S  u.  { 0 } ) )  =  NN0 )
1816, 17eqtr4d 2501 . . . . . . . 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 2457 . . . . . . . . . . . 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 22733 . . . . . . . . . . 11  |-  -.  ( ph  /\  ( `' A " ( S  \  {
0 } ) )  =/=  (/) )
3231pm2.21i 131 . . . . . . . . . 10  |-  ( (
ph  /\  ( `' A " ( S  \  { 0 } ) )  =/=  (/) )  -> 
( `' A "
( S  \  {
0 } ) )  =  (/) )
3320, 32pm2.61dane 2775 . . . . . . . . 9  |-  ( ph  ->  ( `' A "
( S  \  {
0 } ) )  =  (/) )
3433uneq1d 3653 . . . . . . . 8  |-  ( ph  ->  ( ( `' A " ( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )  =  ( (/)  u.  ( `' A " { 0 } ) ) )
35 undif1 3906 . . . . . . . . . 10  |-  ( ( S  \  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
3635imaeq2i 5345 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( `' A " ( S  u.  {
0 } ) )
37 imaundi 5425 . . . . . . . . 9  |-  ( `' A " ( ( S  \  { 0 } )  u.  {
0 } ) )  =  ( ( `' A " ( S 
\  { 0 } ) )  u.  ( `' A " { 0 } ) )
3836, 37eqtr3i 2488 . . . . . . . 8  |-  ( `' A " ( S  u.  { 0 } ) )  =  ( ( `' A "
( S  \  {
0 } ) )  u.  ( `' A " { 0 } ) )
39 un0 3819 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( `' A " { 0 } )
40 uncom 3644 . . . . . . . . 9  |-  ( ( `' A " { 0 } )  u.  (/) )  =  ( (/)  u.  ( `' A " { 0 } ) )
4139, 40eqtr3i 2488 . . . . . . . 8  |-  ( `' A " { 0 } )  =  (
(/)  u.  ( `' A " { 0 } ) )
4234, 38, 413eqtr4g 2523 . . . . . . 7  |-  ( ph  ->  ( `' A "
( S  u.  {
0 } ) )  =  ( `' A " { 0 } ) )
4319, 42eqtrd 2498 . . . . . 6  |-  ( ph  ->  dom  A  =  ( `' A " { 0 } ) )
44 eqimss 3551 . . . . . 6  |-  ( dom 
A  =  ( `' A " { 0 } )  ->  dom  A 
C_  ( `' A " { 0 } ) )
4543, 44syl 16 . . . . 5  |-  ( ph  ->  dom  A  C_  ( `' A " { 0 } ) )
46 ffun 5739 . . . . . . 7  |-  ( A : NN0 --> ( S  u.  { 0 } )  ->  Fun  A )
4712, 46syl 16 . . . . . 6  |-  ( ph  ->  Fun  A )
48 ssid 3518 . . . . . 6  |-  dom  A  C_ 
dom  A
49 funimass3 6004 . . . . . 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 3544 . . 3  |-  ( ph  ->  ran  A  C_  { 0 } )
53 df-f 5598 . . 3  |-  ( A : NN0 --> { 0 }  <->  ( A  Fn  NN0 
/\  ran  A  C_  { 0 } ) )
5414, 52, 53sylanbrc 664 . 2  |-  ( ph  ->  A : NN0 --> { 0 } )
55 c0ex 9607 . . 3  |-  0  e.  _V
5655fconst2 6129 . 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3793   {csn 4032    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697   ^cexp 12169   sum_csu 13520   0pc0p 22202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-0p 22203
This theorem is referenced by:  coeeulem  22747
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