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Theorem coecj 19659
Description: Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
plycj.1  |-  N  =  (deg `  F )
plycj.2  |-  G  =  ( ( *  o.  F )  o.  *
)
coecj.3  |-  A  =  (coeff `  F )
Assertion
Ref Expression
coecj  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  G
)  =  ( *  o.  A ) )

Proof of Theorem coecj
Dummy variables  x  k  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plycj.1 . . 3  |-  N  =  (deg `  F )
2 plycj.2 . . 3  |-  G  =  ( ( *  o.  F )  o.  *
)
3 cjcl 11590 . . . 4  |-  ( x  e.  CC  ->  (
* `  x )  e.  CC )
43adantl 452 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
* `  x )  e.  CC )
5 plyssc 19582 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
65sseli 3176 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
71, 2, 4, 6plycj 19658 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 dgrcl 19615 . . 3  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
91, 8syl5eqel 2367 . 2  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
10 cjf 11589 . . 3  |-  * : CC --> CC
11 coecj.3 . . . 4  |-  A  =  (coeff `  F )
1211coef3 19614 . . 3  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
13 fco 5398 . . 3  |-  ( ( * : CC --> CC  /\  A : NN0 --> CC )  ->  ( *  o.  A ) : NN0 --> CC )
1410, 12, 13sylancr 644 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( *  o.  A ) : NN0 --> CC )
15 fvco3 5596 . . . . . . . . 9  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( *  o.  A ) `  k
)  =  ( * `
 ( A `  k ) ) )
1612, 15sylan 457 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( *  o.  A
) `  k )  =  ( * `  ( A `  k ) ) )
17 cj0 11643 . . . . . . . . . 10  |-  ( * `
 0 )  =  0
1817eqcomi 2287 . . . . . . . . 9  |-  0  =  ( * ` 
0 )
1918a1i 10 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  0  =  ( * ` 
0 ) )
2016, 19eqeq12d 2297 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =  0  <->  (
* `  ( A `  k ) )  =  ( * `  0
) ) )
21 ffvelrn 5663 . . . . . . . . 9  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
2212, 21sylan 457 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
23 0cn 8831 . . . . . . . . 9  |-  0  e.  CC
2423a1i 10 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  0  e.  CC )
25 cj11 11647 . . . . . . . 8  |-  ( ( ( A `  k
)  e.  CC  /\  0  e.  CC )  ->  ( ( * `  ( A `  k ) )  =  ( * `
 0 )  <->  ( A `  k )  =  0 ) )
2622, 24, 25syl2anc 642 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( * `  ( A `  k )
)  =  ( * `
 0 )  <->  ( A `  k )  =  0 ) )
2720, 26bitrd 244 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =  0  <->  ( A `  k )  =  0 ) )
2827necon3bid 2481 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =/=  0  <->  ( A `  k )  =/=  0 ) )
2911, 1dgrub2 19617 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
30 plyco0 19574 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  N ) ) )
319, 12, 30syl2anc 642 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( ( A " ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  N ) ) )
3229, 31mpbid 201 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  N ) )
3332r19.21bi 2641 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
3428, 33sylbid 206 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =/=  0  -> 
k  <_  N )
)
3534ralrimiva 2626 . . 3  |-  ( F  e.  (Poly `  S
)  ->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) )
36 plyco0 19574 . . . 4  |-  ( ( N  e.  NN0  /\  ( *  o.  A
) : NN0 --> CC )  ->  ( ( ( *  o.  A )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) ) )
379, 14, 36syl2anc 642 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( (
( *  o.  A
) " ( ZZ>= `  ( N  +  1
) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) ) )
3835, 37mpbird 223 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( (
*  o.  A )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 } )
391, 2, 11plycjlem 19657 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) ) )
407, 9, 14, 38, 39coeeq 19609 1  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  G
)  =  ( *  o.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {csn 3640   class class class wbr 4023   "cima 4692    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    <_ cle 8868   NN0cn0 9965   ZZ>=cuz 10230   *ccj 11581  Polycply 19566  coeffccoe 19568  degcdgr 19569
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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