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Theorem coecj 22844
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 13023 . . . 4  |-  ( x  e.  CC  ->  (
* `  x )  e.  CC )
43adantl 464 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
* `  x )  e.  CC )
5 plyssc 22766 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
65sseli 3485 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
71, 2, 4, 6plycj 22843 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 dgrcl 22799 . . 3  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
91, 8syl5eqel 2546 . 2  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
10 cjf 13022 . . 3  |-  * : CC --> CC
11 coecj.3 . . . 4  |-  A  =  (coeff `  F )
1211coef3 22798 . . 3  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
13 fco 5723 . . 3  |-  ( ( * : CC --> CC  /\  A : NN0 --> CC )  ->  ( *  o.  A ) : NN0 --> CC )
1410, 12, 13sylancr 661 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( *  o.  A ) : NN0 --> CC )
15 fvco3 5925 . . . . . . . . 9  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( *  o.  A ) `  k
)  =  ( * `
 ( A `  k ) ) )
1612, 15sylan 469 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( *  o.  A
) `  k )  =  ( * `  ( A `  k ) ) )
17 cj0 13076 . . . . . . . . . 10  |-  ( * `
 0 )  =  0
1817eqcomi 2467 . . . . . . . . 9  |-  0  =  ( * ` 
0 )
1918a1i 11 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  0  =  ( * ` 
0 ) )
2016, 19eqeq12d 2476 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =  0  <->  (
* `  ( A `  k ) )  =  ( * `  0
) ) )
2112ffvelrnda 6007 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
22 0cnd 9578 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  0  e.  CC )
23 cj11 13080 . . . . . . . 8  |-  ( ( ( A `  k
)  e.  CC  /\  0  e.  CC )  ->  ( ( * `  ( A `  k ) )  =  ( * `
 0 )  <->  ( A `  k )  =  0 ) )
2421, 22, 23syl2anc 659 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( * `  ( A `  k )
)  =  ( * `
 0 )  <->  ( A `  k )  =  0 ) )
2520, 24bitrd 253 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =  0  <->  ( A `  k )  =  0 ) )
2625necon3bid 2712 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =/=  0  <->  ( A `  k )  =/=  0 ) )
2711, 1dgrub2 22801 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( A " ( ZZ>= `  ( N  +  1 ) ) )  =  { 0 } )
28 plyco0 22758 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A : NN0 --> CC )  ->  ( ( A
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  N ) ) )
299, 12, 28syl2anc 659 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  ( ( A " ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  N ) ) )
3027, 29mpbid 210 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  A. k  e.  NN0  ( ( A `
 k )  =/=  0  ->  k  <_  N ) )
3130r19.21bi 2823 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
3226, 31sylbid 215 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( ( *  o.  A ) `  k
)  =/=  0  -> 
k  <_  N )
)
3332ralrimiva 2868 . . 3  |-  ( F  e.  (Poly `  S
)  ->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) )
34 plyco0 22758 . . . 4  |-  ( ( N  e.  NN0  /\  ( *  o.  A
) : NN0 --> CC )  ->  ( ( ( *  o.  A )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 }  <->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) ) )
359, 14, 34syl2anc 659 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( (
( *  o.  A
) " ( ZZ>= `  ( N  +  1
) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( *  o.  A ) `
 k )  =/=  0  ->  k  <_  N ) ) )
3633, 35mpbird 232 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( (
*  o.  A )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 } )
371, 2, 11plycjlem 22842 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) ) )
387, 9, 14, 36, 37coeeq 22793 1  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  G
)  =  ( *  o.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {csn 4016   class class class wbr 4439   "cima 4991    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618   NN0cn0 10791   ZZ>=cuz 11082   *ccj 13014  Polycply 22750  coeffccoe 22752  degcdgr 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-rlim 13397  df-sum 13594  df-0p 22246  df-ply 22754  df-coe 22756  df-dgr 22757
This theorem is referenced by: (None)
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