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Theorem plycj 20200
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 2438 . . . . 5  |-  (coeff `  F )  =  (coeff `  F )
52, 3, 4plycjlem 20199 . . . 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 20118 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
81, 7syl 16 . . . . 5  |-  ( ph  ->  S  C_  CC )
9 0cn 9089 . . . . . . 7  |-  0  e.  CC
109a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  CC )
1110snssd 3945 . . . . 5  |-  ( ph  ->  { 0 }  C_  CC )
128, 11unssd 3525 . . . 4  |-  ( ph  ->  ( S  u.  {
0 } )  C_  CC )
13 dgrcl 20157 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
141, 13syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
152, 14syl5eqel 2522 . . . 4  |-  ( ph  ->  N  e.  NN0 )
164coef 20154 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> ( S  u.  { 0 } ) )
171, 16syl 16 . . . . . 6  |-  ( ph  ->  (coeff `  F ) : NN0 --> ( S  u.  { 0 } ) )
18 elfznn0 11088 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
19 fvco3 5803 . . . . . 6  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( ( *  o.  (coeff `  F )
) `  k )  =  ( * `  ( (coeff `  F ) `  k ) ) )
2017, 18, 19syl2an 465 . . . . 5  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  =  ( * `  ( (coeff `  F ) `  k
) ) )
21 ffvelrn 5871 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ( S  u.  { 0 } )  /\  k  e.  NN0 )  -> 
( (coeff `  F
) `  k )  e.  ( S  u.  {
0 } ) )
2217, 18, 21syl2an 465 . . . . . 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 2791 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  S  ( * `  x
)  e.  S )
25 fveq2 5731 . . . . . . . . . . . 12  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( * `  x )  =  ( * `  ( (coeff `  F ) `  k
) ) )
2625eleq1d 2504 . . . . . . . . . . 11  |-  ( x  =  ( (coeff `  F ) `  k
)  ->  ( (
* `  x )  e.  S  <->  ( * `  ( (coeff `  F ) `  k ) )  e.  S ) )
2726rspccv 3051 . . . . . . . . . 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 3840 . . . . . . . . . . . . 13  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
(coeff `  F ) `  k )  =  0 )
3029fveq2d 5735 . . . . . . . . . . . 12  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  ( * `  0 ) )
31 cj0 11968 . . . . . . . . . . . 12  |-  ( * `
 0 )  =  0
3230, 31syl6eq 2486 . . . . . . . . . . 11  |-  ( ( (coeff `  F ) `  k )  e.  {
0 }  ->  (
* `  ( (coeff `  F ) `  k
) )  =  0 )
33 fvex 5745 . . . . . . . . . . . 12  |-  ( * `
 ( (coeff `  F ) `  k
) )  e.  _V
3433elsnc 3839 . . . . . . . . . . 11  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  {
0 }  <->  ( * `  ( (coeff `  F
) `  k )
)  =  0 )
3532, 34sylibr 205 . . . . . . . . . 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 813 . . . . . . . 8  |-  ( ph  ->  ( ( ( (coeff `  F ) `  k
)  e.  S  \/  ( (coeff `  F ) `  k )  e.  {
0 } )  -> 
( ( * `  ( (coeff `  F ) `  k ) )  e.  S  \/  ( * `
 ( (coeff `  F ) `  k
) )  e.  {
0 } ) ) )
38 elun 3490 . . . . . . . 8  |-  ( ( (coeff `  F ) `  k )  e.  ( S  u.  { 0 } )  <->  ( (
(coeff `  F ) `  k )  e.  S  \/  ( (coeff `  F
) `  k )  e.  { 0 } ) )
39 elun 3490 . . . . . . . 8  |-  ( ( * `  ( (coeff `  F ) `  k
) )  e.  ( S  u.  { 0 } )  <->  ( (
* `  ( (coeff `  F ) `  k
) )  e.  S  \/  ( * `  (
(coeff `  F ) `  k ) )  e. 
{ 0 } ) )
4037, 38, 393imtr4g 263 . . . . . . 7  |-  ( ph  ->  ( ( (coeff `  F ) `  k
)  e.  ( S  u.  { 0 } )  ->  ( * `  ( (coeff `  F
) `  k )
)  e.  ( S  u.  { 0 } ) ) )
4140adantr 453 . . . . . 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 2512 . . . 4  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  (coeff `  F ) ) `  k )  e.  ( S  u.  { 0 } ) )
4412, 15, 43elplyd 20126 . . 3  |-  ( ph  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  (coeff `  F )
) `  k )  x.  ( z ^ k
) ) )  e.  (Poly `  ( S  u.  { 0 } ) ) )
456, 44eqeltrd 2512 . 2  |-  ( ph  ->  G  e.  (Poly `  ( S  u.  { 0 } ) ) )
46 plyun0 20121 . 2  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
4745, 46syl6eleq 2528 1  |-  ( ph  ->  G  e.  (Poly `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    u. cun 3320    C_ wss 3322   {csn 3816    e. cmpt 4269    o. ccom 4885   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995    x. cmul 9000   NN0cn0 10226   ...cfz 11048   ^cexp 11387   *ccj 11906   sum_csu 12484  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem is referenced by:  coecj  20201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115
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