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Theorem plycj 22403
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 2462 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
52, 3, 4plycjlem 22402 . . . 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 22321 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
81, 7syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
9 0cnd 9580 . . . . . 6  |-  ( ph  ->  0  e.  CC )
109snssd 4167 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
118, 10unssd 3675 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
12 dgrcl 22360 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
131, 12syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
142, 13syl5eqel 2554 . . . 4  |-  ( ph  ->  N  e.  NN0 )
154coef 22357 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
161, 15syl 16 . . . . . 6  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
17 elfznn0 11761 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
18 fvco3 5937 . . . . . 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 6012 . . . . . . 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 2873 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( * `  x
)  e.  S )
24 fveq2 5859 . . . . . . . . . . . 12  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( * `  x )  =  ( * `  ( (coeff `  F ) `  k
) ) )
2524eleq1d 2531 . . . . . . . . . . 11  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( (
* `  x )  e.  S  <->  ( * `  ( (coeff `  F ) `  k ) )  e.  S ) )
2625rspccv 3206 . . . . . . . . . 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 4047 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
(coeff `  F ) `  k )  =  0 )
2928fveq2d 5863 . . . . . . . . . . . 12  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  ( * `  0 ) )
30 cj0 12943 . . . . . . . . . . . 12  |-  ( * `
 0 )  =  0
3129, 30syl6eq 2519 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  0 )
32 fvex 5869 . . . . . . . . . . . 12  |-  ( * `
 ( (coeff `  F ) `  k
) )  e.  _V
3332elsnc 4046 . . . . . . . . . . 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 835 . . . . . . . 8  |-  ( ph  ->  ( ( ( (coeff `  F ) `  k
)  e.  S  \/  ( (coeff `  F ) `  k )  e.  {
0 } )  -> 
( ( * `  ( (coeff `  F ) `  k ) )  e.  S  \/  ( * `
 ( (coeff `  F ) `  k
) )  e.  {
0 } ) ) )
37 elun 3640 . . . . . . . 8  |-  ( ( (coeff `  F ) `  k )  e.  ( S  u.  { 0 } )  <->  ( (
(coeff `  F ) `  k )  e.  S  \/  ( (coeff `  F
) `  k )  e.  { 0 } ) )
38 elun 3640 . . . . . . . 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 2550 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  e.  ( S  u.  { 0 } ) )
4311, 14, 42elplyd 22329 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
446, 43eqeltrd 2550 . 2  |-  ( ph  ->  G  e.  (Poly `  ( S  u.  { 0 } ) ) )
45 plyun0 22324 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4644, 45syl6eleq 2560 1  |-  ( ph  ->  G  e.  (Poly `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809    u. cun 3469    C_ wss 3471   {csn 4022    |-> cmpt 4500    o. ccom 4998   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481   0cc0 9483    x. cmul 9488   NN0cn0 10786   ...cfz 11663   ^cexp 12124   *ccj 12881   sum_csu 13459  Polycply 22311  coeffccoe 22313  degcdgr 22314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-sum 13460  df-0p 21807  df-ply 22315  df-coe 22317  df-dgr 22318
This theorem is referenced by:  coecj  22404
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