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Theorem plymulx 29486
Description: Coefficients of a polynomial multiplyed by  Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx  |-  ( F  e.  (Poly `  RR )  ->  (coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
Distinct variable group:    n, F

Proof of Theorem plymulx
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 ax-resscn 9622 . . . . . . 7  |-  RR  C_  CC
2 1re 9668 . . . . . . 7  |-  1  e.  RR
3 plyid 23212 . . . . . . 7  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
41, 2, 3mp2an 683 . . . . . 6  |-  Xp  e.  (Poly `  RR )
5 plymul02 29484 . . . . . . 7  |-  ( Xp  e.  (Poly `  RR )  ->  ( 0p  oF  x.  Xp )  =  0p )
65fveq2d 5892 . . . . . 6  |-  ( Xp  e.  (Poly `  RR )  ->  (coeff `  ( 0p  oF  x.  Xp ) )  =  (coeff `  0p ) )
74, 6ax-mp 5 . . . . 5  |-  (coeff `  ( 0p  oF  x.  Xp ) )  =  (coeff `  0p )
8 fconstmpt 4897 . . . . . 6  |-  ( NN0 
X.  { 0 } )  =  ( n  e.  NN0  |->  0 )
9 coe0 23259 . . . . . 6  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
10 eqidd 2463 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  n  =  0 )  ->  0  =  0 )
11 elnnne0 10912 . . . . . . . . . . 11  |-  ( n  e.  NN  <->  ( n  e.  NN0  /\  n  =/=  0 ) )
12 df-ne 2635 . . . . . . . . . . . 12  |-  ( n  =/=  0  <->  -.  n  =  0 )
1312anbi2i 705 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  /\  n  =/=  0 )  <->  ( n  e.  NN0  /\  -.  n  =  0 ) )
1411, 13bitr2i 258 . . . . . . . . . 10  |-  ( ( n  e.  NN0  /\  -.  n  =  0
)  <->  n  e.  NN )
15 nnm1nn0 10940 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
1614, 15sylbi 200 . . . . . . . . 9  |-  ( ( n  e.  NN0  /\  -.  n  =  0
)  ->  ( n  -  1 )  e. 
NN0 )
17 eqidd 2463 . . . . . . . . . 10  |-  ( m  =  ( n  - 
1 )  ->  0  =  0 )
18 fconstmpt 4897 . . . . . . . . . . 11  |-  ( NN0 
X.  { 0 } )  =  ( m  e.  NN0  |->  0 )
199, 18eqtri 2484 . . . . . . . . . 10  |-  (coeff ` 
0p )  =  ( m  e.  NN0  |->  0 )
20 c0ex 9663 . . . . . . . . . 10  |-  0  e.  _V
2117, 19, 20fvmpt 5971 . . . . . . . . 9  |-  ( ( n  -  1 )  e.  NN0  ->  ( (coeff `  0p ) `
 ( n  - 
1 ) )  =  0 )
2216, 21syl 17 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  -.  n  =  0
)  ->  ( (coeff `  0p ) `  ( n  -  1
) )  =  0 )
2310, 22ifeqda 3926 . . . . . . 7  |-  ( n  e.  NN0  ->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) )  =  0 )
2423mpteq2ia 4499 . . . . . 6  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) ) )  =  ( n  e.  NN0  |->  0 )
258, 9, 243eqtr4ri 2495 . . . . 5  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) ) )  =  (coeff ` 
0p )
267, 25eqtr4i 2487 . . . 4  |-  (coeff `  ( 0p  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) ) )
27 oveq1 6322 . . . . 5  |-  ( F  =  0p  -> 
( F  oF  x.  Xp )  =  ( 0p  oF  x.  Xp ) )
2827fveq2d 5892 . . . 4  |-  ( F  =  0p  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( 0p  oF  x.  Xp ) ) )
29 simpl 463 . . . . . . . 8  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  F  =  0p )
3029fveq2d 5892 . . . . . . 7  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  (coeff `  F
)  =  (coeff ` 
0p ) )
3130fveq1d 5890 . . . . . 6  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  =  ( (coeff `  0p ) `
 ( n  - 
1 ) ) )
3231ifeq2d 3912 . . . . 5  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) )  =  if ( n  =  0 ,  0 ,  ( (coeff `  0p
) `  ( n  -  1 ) ) ) )
3332mpteq2dva 4503 . . . 4  |-  ( F  =  0p  -> 
( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) )  =  ( n  e. 
NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff ` 
0p ) `  ( n  -  1
) ) ) ) )
3426, 28, 333eqtr4a 2522 . . 3  |-  ( F  =  0p  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
3534adantl 472 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  F  =  0p )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
36 simpl 463 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  F  e.  (Poly `  RR ) )
37 elsncg 4003 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( F  e. 
{ 0p }  <->  F  =  0p ) )
3837notbid 300 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( -.  F  e.  { 0p }  <->  -.  F  =  0p ) )
3938biimpar 492 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  -.  F  e.  { 0p } )
4036, 39eldifd 3427 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  F  e.  ( (Poly `  RR )  \  { 0p }
) )
41 plymulx0 29485 . . 3  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
4240, 41syl 17 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  (coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
4335, 42pm2.61dan 805 1  |-  ( F  e.  (Poly `  RR )  ->  (coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633    \ cdif 3413    C_ wss 3416   ifcif 3893   {csn 3980    |-> cmpt 4475    X. cxp 4851   ` cfv 5601  (class class class)co 6315    oFcof 6556   CCcc 9563   RRcr 9564   0cc0 9565   1c1 9566    x. cmul 9570    - cmin 9886   NNcn 10637   NN0cn0 10898   0pc0p 22676  Polycply 23187   Xpcidp 23188  coeffccoe 23189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-pm 7501  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-fz 11814  df-fzo 11947  df-fl 12060  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-clim 13601  df-rlim 13602  df-sum 13802  df-0p 22677  df-ply 23191  df-idp 23192  df-coe 23193  df-dgr 23194
This theorem is referenced by: (None)
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