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Theorem plycj 20156
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 2412 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
52, 3, 4plycjlem 20155 . . . 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 20074 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
81, 7syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
9 0cn 9048 . . . . . . 7  |-  0  e.  CC
109a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  CC )
1110snssd 3911 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
128, 11unssd 3491 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
13 dgrcl 20113 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
152, 14syl5eqel 2496 . . . 4  |-  ( ph  ->  N  e.  NN0 )
164coef 20110 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
171, 16syl 16 . . . . . 6  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
18 elfznn0 11047 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
19 fvco3 5767 . . . . . 6  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( ( *  o.  (coeff `  F )
) `  k )  =  ( * `  ( (coeff `  F ) `  k ) ) )
2017, 18, 19syl2an 464 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  =  ( * `  ( (coeff `  F ) `  k
) ) )
21 ffvelrn 5835 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } ) )
2217, 18, 21syl2an 464 . . . . . 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 2757 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( * `  x
)  e.  S )
25 fveq2 5695 . . . . . . . . . . . 12  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( * `  x )  =  ( * `  ( (coeff `  F ) `  k
) ) )
2625eleq1d 2478 . . . . . . . . . . 11  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( (
* `  x )  e.  S  <->  ( * `  ( (coeff `  F ) `  k ) )  e.  S ) )
2726rspccv 3017 . . . . . . . . . 10  |-  ( A. x  e.  S  (
* `  x )  e.  S  ->  ( ( (coeff `  F ) `  k )  e.  S  ->  ( * `  (
(coeff `  F ) `  k ) )  e.  S ) )
2824, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  S  -> 
( * `  (
(coeff `  F ) `  k ) )  e.  S ) )
29 elsni 3806 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
(coeff `  F ) `  k )  =  0 )
3029fveq2d 5699 . . . . . . . . . . . 12  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  ( * `  0 ) )
31 cj0 11926 . . . . . . . . . . . 12  |-  ( * `
 0 )  =  0
3230, 31syl6eq 2460 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  0 )
33 fvex 5709 . . . . . . . . . . . 12  |-  ( * `
 ( (coeff `  F ) `  k
) )  e.  _V
3433elsnc 3805 . . . . . . . . . . 11  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  {
0 }  <->  ( * `  ( (coeff `  F
) `  k )
)  =  0 )
3532, 34sylibr 204 . . . . . . . . . 10  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  e.  {
0 } )
3635a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  { 0 }  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  { 0 } ) )
3728, 36orim12d 812 . . . . . . . 8  |-  ( ph  ->  ( ( ( (coeff `  F ) `  k
)  e.  S  \/  ( (coeff `  F ) `  k )  e.  {
0 } )  -> 
( ( * `  ( (coeff `  F ) `  k ) )  e.  S  \/  ( * `
 ( (coeff `  F ) `  k
) )  e.  {
0 } ) ) )
38 elun 3456 . . . . . . . 8  |-  ( ( (coeff `  F ) `  k )  e.  ( S  u.  { 0 } )  <->  ( (
(coeff `  F ) `  k )  e.  S  \/  ( (coeff `  F
) `  k )  e.  { 0 } ) )
39 elun 3456 . . . . . . . 8  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } )  <->  ( (
* `  ( (coeff `  F ) `  k
) )  e.  S  \/  ( * `  (
(coeff `  F ) `  k ) )  e. 
{ 0 } ) )
4037, 38, 393imtr4g 262 . . . . . . 7  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  ( S  u.  { 0 } )  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  ( S  u.  { 0 } ) ) )
4140adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } )  -> 
( * `  (
(coeff `  F ) `  k ) )  e.  ( S  u.  {
0 } ) ) )
4222, 41mpd 15 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } ) )
4320, 42eqeltrd 2486 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  e.  ( S  u.  { 0 } ) )
4412, 15, 43elplyd 20082 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
456, 44eqeltrd 2486 . 2  |-  ( ph  ->  G  e.  (Poly `  ( S  u.  { 0 } ) ) )
46 plyun0 20077 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4745, 46syl6eleq 2502 1  |-  ( ph  ->  G  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674    u. cun 3286    C_ wss 3288   {csn 3782    e. cmpt 4234    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954    x. cmul 8959   NN0cn0 10185   ...cfz 11007   ^cexp 11345   *ccj 11864   sum_csu 12442  Polycply 20064  coeffccoe 20066  degcdgr 20067
This theorem is referenced by:  coecj  20157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-0p 19523  df-ply 20068  df-coe 20070  df-dgr 20071
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