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Theorem coecj 19675
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 11606 . . . 4  |-  ( x  e.  CC  ->  (
* `  x )  e.  CC )
43adantl 452 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
* `  x )  e.  CC )
5 plyssc 19598 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
65sseli 3189 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
71, 2, 4, 6plycj 19674 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 dgrcl 19631 . . 3  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
91, 8syl5eqel 2380 . 2  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
10 cjf 11605 . . 3  |-  * : CC --> CC
11 coecj.3 . . . 4  |-  A  =  (coeff `  F )
1211coef3 19630 . . 3  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
13 fco 5414 . . 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 5612 . . . . . . . . 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 11659 . . . . . . . . . 10  |-  ( * `
 0 )  =  0
1817eqcomi 2300 . . . . . . . . 9  |-  0  =  ( * ` 
0 )
1918a1i 10 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  0  =  ( * ` 
0 ) )
2016, 19eqeq12d 2310 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =  0  <->  (
* `  ( A `  k ) )  =  ( * `  0
) ) )
21 ffvelrn 5679 . . . . . . . . 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 8847 . . . . . . . . 9  |-  0  e.  CC
2423a1i 10 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  0  e.  CC )
25 cj11 11663 . . . . . . . 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 2494 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =/=  0  <->  ( A `  k )  =/=  0 ) )
2911, 1dgrub2 19633 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
30 plyco0 19590 . . . . . . . 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 2654 . . . . 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 2639 . . 3  |-  ( F  e.  (Poly `  S
)  ->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) )
36 plyco0 19590 . . . 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 19673 . 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 19625 1  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  G
)  =  ( *  o.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {csn 3653   class class class wbr 4039   "cima 4708    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884   NN0cn0 9981   ZZ>=cuz 10246   *ccj 11597  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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