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Theorem plyf 22721
Description: The polynomial is a function on the complex numbers. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plyf  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )

Proof of Theorem plyf
Dummy variables  k 
a  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elply 22718 . . 3  |-  ( F  e.  (Poly `  S
)  <->  ( S  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
21simprbi 464 . 2  |-  ( F  e.  (Poly `  S
)  ->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
3 fzfid 12086 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  ( 0 ... n
)  e.  Fin )
4 plybss 22717 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
5 0cnd 9606 . . . . . . . . . . . 12  |-  ( F  e.  (Poly `  S
)  ->  0  e.  CC )
65snssd 4177 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  { 0 }  C_  CC )
74, 6unssd 3676 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  ( S  u.  { 0 } ) 
C_  CC )
87ad2antrr 725 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  ( S  u.  {
0 } )  C_  CC )
98adantr 465 . . . . . . . 8  |-  ( ( ( ( F  e.  (Poly `  S )  /\  ( n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  /\  k  e.  (
0 ... n ) )  ->  ( S  u.  { 0 } )  C_  CC )
10 simplrr 762 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) )
11 cnex 9590 . . . . . . . . . . . 12  |-  CC  e.  _V
12 ssexg 4602 . . . . . . . . . . . 12  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  CC  e.  _V )  ->  ( S  u.  { 0 } )  e. 
_V )
138, 11, 12sylancl 662 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  ( S  u.  {
0 } )  e. 
_V )
14 nn0ex 10822 . . . . . . . . . . 11  |-  NN0  e.  _V
15 elmapg 7451 . . . . . . . . . . 11  |-  ( ( ( S  u.  {
0 } )  e. 
_V  /\  NN0  e.  _V )  ->  ( a  e.  ( ( S  u.  { 0 } )  ^m  NN0 )  <->  a : NN0 --> ( S  u.  { 0 } ) ) )
1613, 14, 15sylancl 662 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  ( a  e.  ( ( S  u.  {
0 } )  ^m  NN0 )  <->  a : NN0 --> ( S  u.  { 0 } ) ) )
1710, 16mpbid 210 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  a : NN0 --> ( S  u.  { 0 } ) )
18 elfznn0 11797 . . . . . . . . 9  |-  ( k  e.  ( 0 ... n )  ->  k  e.  NN0 )
19 ffvelrn 6030 . . . . . . . . 9  |-  ( ( a : NN0 --> ( S  u.  { 0 } )  /\  k  e. 
NN0 )  ->  (
a `  k )  e.  ( S  u.  {
0 } ) )
2017, 18, 19syl2an 477 . . . . . . . 8  |-  ( ( ( ( F  e.  (Poly `  S )  /\  ( n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  /\  k  e.  (
0 ... n ) )  ->  ( a `  k )  e.  ( S  u.  { 0 } ) )
219, 20sseldd 3500 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  ( n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  /\  k  e.  (
0 ... n ) )  ->  ( a `  k )  e.  CC )
22 simpr 461 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  ->  z  e.  CC )
23 expcl 12187 . . . . . . . 8  |-  ( ( z  e.  CC  /\  k  e.  NN0 )  -> 
( z ^ k
)  e.  CC )
2422, 18, 23syl2an 477 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  ( n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  /\  k  e.  (
0 ... n ) )  ->  ( z ^
k )  e.  CC )
2521, 24mulcld 9633 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  ( n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  /\  k  e.  (
0 ... n ) )  ->  ( ( a `
 k )  x.  ( z ^ k
) )  e.  CC )
263, 25fsumcl 13567 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  /\  z  e.  CC )  -> 
sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) )  e.  CC )
27 eqid 2457 . . . . 5  |-  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n
) ( ( a `
 k )  x.  ( z ^ k
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )
2826, 27fmptd 6056 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) : CC --> CC )
29 feq1 5719 . . . 4  |-  ( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  ->  ( F : CC --> CC  <->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) : CC --> CC ) )
3028, 29syl5ibrcom 222 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  (
n  e.  NN0  /\  a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) ) )  -> 
( F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  ->  F : CC
--> CC ) )
3130rexlimdvva 2956 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  ->  F : CC
--> CC ) )
322, 31mpd 15 1  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    u. cun 3469    C_ wss 3471   {csn 4032    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   CCcc 9507   0cc0 9509    x. cmul 9514   NN0cn0 10816   ...cfz 11697   ^cexp 12169   sum_csu 13520  Polycply 22707
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
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-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-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-sum 13521  df-ply 22711
This theorem is referenced by:  plysub  22742  plyco  22764  0dgrb  22769  coe0  22779  coesub  22780  dgrsub  22795  dgrcolem1  22796  dgrcolem2  22797  dgrco  22798  plymul0or  22803  plyreres  22805  dvply2g  22807  dvnply2  22809  plycpn  22811  plydivlem3  22817  plydivlem4  22818  plydiveu  22820  plyremlem  22826  plyrem  22827  facth  22828  fta1lem  22829  fta1  22830  quotcan  22831  vieta1lem1  22832  vieta1lem2  22833  vieta1  22834  plyexmo  22835  elaa  22838  elqaalem3  22843  aannenlem1  22850  aalioulem2  22855  aalioulem3  22856  aalioulem4  22857  taylthlem2  22895  ftalem2  23473  ftalem3  23474  ftalem4  23475  ftalem5  23476  ftalem7  23478  basellem4  23483  basellem5  23484  plymul02  28700  plymulx0  28701  signsplypnf  28704  signsply0  28705  mpaaeu  31282  rngunsnply  31305
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