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Theorem plycj 19658
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 2283 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
52, 3, 4plycjlem 19657 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) ) )
61, 5syl 15 . . 3  |-  ( ph  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( ( *  o.  (coeff `  F ) ) `  k )  x.  (
z ^ k ) ) ) )
7 plybss 19576 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
81, 7syl 15 . . . . 5  |-  ( ph  ->  S  C_  CC )
9 0cn 8831 . . . . . . 7  |-  0  e.  CC
109a1i 10 . . . . . 6  |-  ( ph  ->  0  e.  CC )
1110snssd 3760 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
128, 11unssd 3351 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
13 dgrcl 19615 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 15 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
152, 14syl5eqel 2367 . . . 4  |-  ( ph  ->  N  e.  NN0 )
164coef 19612 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
171, 16syl 15 . . . . . 6  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
18 elfznn0 10822 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
19 fvco3 5596 . . . . . 6  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( ( *  o.  (coeff `  F )
) `  k )  =  ( * `  ( (coeff `  F ) `  k ) ) )
2017, 18, 19syl2an 463 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  =  ( * `  ( (coeff `  F ) `  k
) ) )
21 ffvelrn 5663 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } ) )
2217, 18, 21syl2an 463 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
(coeff `  F ) `  k )  e.  ( S  u.  { 0 } ) )
23 plycj.3 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  (
* `  x )  e.  S )
2423ralrimiva 2626 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( * `  x
)  e.  S )
25 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( * `  x )  =  ( * `  ( (coeff `  F ) `  k
) ) )
2625eleq1d 2349 . . . . . . . . . . 11  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( (
* `  x )  e.  S  <->  ( * `  ( (coeff `  F ) `  k ) )  e.  S ) )
2726rspccv 2881 . . . . . . . . . 10  |-  ( A. x  e.  S  (
* `  x )  e.  S  ->  ( ( (coeff `  F ) `  k )  e.  S  ->  ( * `  (
(coeff `  F ) `  k ) )  e.  S ) )
2824, 27syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  S  -> 
( * `  (
(coeff `  F ) `  k ) )  e.  S ) )
29 elsni 3664 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
(coeff `  F ) `  k )  =  0 )
3029fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  ( * `  0 ) )
31 cj0 11643 . . . . . . . . . . . 12  |-  ( * `
 0 )  =  0
3230, 31syl6eq 2331 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  0 )
33 fvex 5539 . . . . . . . . . . . 12  |-  ( * `
 ( (coeff `  F ) `  k
) )  e.  _V
3433elsnc 3663 . . . . . . . . . . 11  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  {
0 }  <->  ( * `  ( (coeff `  F
) `  k )
)  =  0 )
3532, 34sylibr 203 . . . . . . . . . 10  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  e.  {
0 } )
3635a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  { 0 }  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  { 0 } ) )
3728, 36orim12d 811 . . . . . . . 8  |-  ( ph  ->  ( ( ( (coeff `  F ) `  k
)  e.  S  \/  ( (coeff `  F ) `  k )  e.  {
0 } )  -> 
( ( * `  ( (coeff `  F ) `  k ) )  e.  S  \/  ( * `
 ( (coeff `  F ) `  k
) )  e.  {
0 } ) ) )
38 elun 3316 . . . . . . . 8  |-  ( ( (coeff `  F ) `  k )  e.  ( S  u.  { 0 } )  <->  ( (
(coeff `  F ) `  k )  e.  S  \/  ( (coeff `  F
) `  k )  e.  { 0 } ) )
39 elun 3316 . . . . . . . 8  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } )  <->  ( (
* `  ( (coeff `  F ) `  k
) )  e.  S  \/  ( * `  (
(coeff `  F ) `  k ) )  e. 
{ 0 } ) )
4037, 38, 393imtr4g 261 . . . . . . 7  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  ( S  u.  { 0 } )  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  ( S  u.  { 0 } ) ) )
4140adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } )  -> 
( * `  (
(coeff `  F ) `  k ) )  e.  ( S  u.  {
0 } ) ) )
4222, 41mpd 14 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } ) )
4320, 42eqeltrd 2357 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  e.  ( S  u.  { 0 } ) )
4412, 15, 43elplyd 19584 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
456, 44eqeltrd 2357 . 2  |-  ( ph  ->  G  e.  (Poly `  ( S  u.  { 0 } ) ) )
46 plyun0 19579 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4745, 46syl6eleq 2373 1  |-  ( ph  ->  G  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    C_ wss 3152   {csn 3640    e. cmpt 4077    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742   NN0cn0 9965   ...cfz 10782   ^cexp 11104   *ccj 11581   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  coecj  19659
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-of 6078  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  df-ply 19570  df-coe 19572  df-dgr 19573
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