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Theorem plycj 22800
Description: The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on  ( * `  z ) independently of  z.) (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
plycj.1  |-  N  =  (deg `  F )
plycj.2  |-  G  =  ( ( *  o.  F )  o.  *
)
plycj.3  |-  ( (
ph  /\  x  e.  S )  ->  (
* `  x )  e.  S )
plycj.4  |-  ( ph  ->  F  e.  (Poly `  S ) )
Assertion
Ref Expression
plycj  |-  ( ph  ->  G  e.  (Poly `  S ) )
Distinct variable groups:    x, F    x, N    ph, x    x, S
Allowed substitution hint:    G( x)

Proof of Theorem plycj
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plycj.4 . . . 4  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 plycj.1 . . . . 5  |-  N  =  (deg `  F )
3 plycj.2 . . . . 5  |-  G  =  ( ( *  o.  F )  o.  *
)
4 eqid 2457 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
52, 3, 4plycjlem 22799 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) ) )
61, 5syl 16 . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( *  o.  (coeff `  F ) ) `  k )  x.  (
z ^ k ) ) ) )
7 plybss 22717 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
81, 7syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
9 0cnd 9606 . . . . . 6  |-  ( ph  ->  0  e.  CC )
109snssd 4177 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
118, 10unssd 3676 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
12 dgrcl 22756 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
131, 12syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
142, 13syl5eqel 2549 . . . 4  |-  ( ph  ->  N  e.  NN0 )
154coef 22753 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
161, 15syl 16 . . . . . 6  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
17 elfznn0 11797 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
18 fvco3 5950 . . . . . 6  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( ( *  o.  (coeff `  F )
) `  k )  =  ( * `  ( (coeff `  F ) `  k ) ) )
1916, 17, 18syl2an 477 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  =  ( * `  ( (coeff `  F ) `  k
) ) )
20 ffvelrn 6030 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } ) )
2116, 17, 20syl2an 477 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
(coeff `  F ) `  k )  e.  ( S  u.  { 0 } ) )
22 plycj.3 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
* `  x )  e.  S )
2322ralrimiva 2871 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( * `  x
)  e.  S )
24 fveq2 5872 . . . . . . . . . . . 12  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( * `  x )  =  ( * `  ( (coeff `  F ) `  k
) ) )
2524eleq1d 2526 . . . . . . . . . . 11  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( (
* `  x )  e.  S  <->  ( * `  ( (coeff `  F ) `  k ) )  e.  S ) )
2625rspccv 3207 . . . . . . . . . 10  |-  ( A. x  e.  S  (
* `  x )  e.  S  ->  ( ( (coeff `  F ) `  k )  e.  S  ->  ( * `  (
(coeff `  F ) `  k ) )  e.  S ) )
2723, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  S  -> 
( * `  (
(coeff `  F ) `  k ) )  e.  S ) )
28 elsni 4057 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
(coeff `  F ) `  k )  =  0 )
2928fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  ( * `  0 ) )
30 cj0 13003 . . . . . . . . . . . 12  |-  ( * `
 0 )  =  0
3129, 30syl6eq 2514 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  0 )
32 fvex 5882 . . . . . . . . . . . 12  |-  ( * `
 ( (coeff `  F ) `  k
) )  e.  _V
3332elsnc 4056 . . . . . . . . . . 11  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  {
0 }  <->  ( * `  ( (coeff `  F
) `  k )
)  =  0 )
3431, 33sylibr 212 . . . . . . . . . 10  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  e.  {
0 } )
3534a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  { 0 }  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  { 0 } ) )
3627, 35orim12d 838 . . . . . . . 8  |-  ( ph  ->  ( ( ( (coeff `  F ) `  k
)  e.  S  \/  ( (coeff `  F ) `  k )  e.  {
0 } )  -> 
( ( * `  ( (coeff `  F ) `  k ) )  e.  S  \/  ( * `
 ( (coeff `  F ) `  k
) )  e.  {
0 } ) ) )
37 elun 3641 . . . . . . . 8  |-  ( ( (coeff `  F ) `  k )  e.  ( S  u.  { 0 } )  <->  ( (
(coeff `  F ) `  k )  e.  S  \/  ( (coeff `  F
) `  k )  e.  { 0 } ) )
38 elun 3641 . . . . . . . 8  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } )  <->  ( (
* `  ( (coeff `  F ) `  k
) )  e.  S  \/  ( * `  (
(coeff `  F ) `  k ) )  e. 
{ 0 } ) )
3936, 37, 383imtr4g 270 . . . . . . 7  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  ( S  u.  { 0 } )  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  ( S  u.  { 0 } ) ) )
4039adantr 465 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } )  -> 
( * `  (
(coeff `  F ) `  k ) )  e.  ( S  u.  {
0 } ) ) )
4121, 40mpd 15 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } ) )
4219, 41eqeltrd 2545 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  e.  ( S  u.  { 0 } ) )
4311, 14, 42elplyd 22725 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
446, 43eqeltrd 2545 . 2  |-  ( ph  ->  G  e.  (Poly `  ( S  u.  { 0 } ) ) )
45 plyun0 22720 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4644, 45syl6eleq 2555 1  |-  ( ph  ->  G  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    u. cun 3469    C_ wss 3471   {csn 4032    |-> cmpt 4515    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509    x. cmul 9514   NN0cn0 10816   ...cfz 11697   ^cexp 12169   *ccj 12941   sum_csu 13520  Polycply 22707  coeffccoe 22709  degcdgr 22710
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-of 6539  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  df-ply 22711  df-coe 22713  df-dgr 22714
This theorem is referenced by:  coecj  22801
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