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Theorem plymulx 28688
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 9460 . . . . . . 7  |-  RR  C_  CC
2 1re 9506 . . . . . . 7  |-  1  e.  RR
3 plyid 22691 . . . . . . 7  |-  ( ( RR  C_  CC  /\  1  e.  RR )  ->  Xp  e.  (Poly `  RR ) )
41, 2, 3mp2an 670 . . . . . 6  |-  Xp  e.  (Poly `  RR )
5 plymul02 28686 . . . . . . 7  |-  ( Xp  e.  (Poly `  RR )  ->  ( 0p  oF  x.  Xp )  =  0p )
65fveq2d 5778 . . . . . 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 4957 . . . . . 6  |-  ( NN0 
X.  { 0 } )  =  ( n  e.  NN0  |->  0 )
9 coe0 22738 . . . . . 6  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
10 eqidd 2383 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  n  =  0 )  ->  0  =  0 )
11 elnnne0 10726 . . . . . . . . . . 11  |-  ( n  e.  NN  <->  ( n  e.  NN0  /\  n  =/=  0 ) )
12 df-ne 2579 . . . . . . . . . . . 12  |-  ( n  =/=  0  <->  -.  n  =  0 )
1312anbi2i 692 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  /\  n  =/=  0 )  <->  ( n  e.  NN0  /\  -.  n  =  0 ) )
1411, 13bitr2i 250 . . . . . . . . . 10  |-  ( ( n  e.  NN0  /\  -.  n  =  0
)  <->  n  e.  NN )
15 nnm1nn0 10754 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
1614, 15sylbi 195 . . . . . . . . 9  |-  ( ( n  e.  NN0  /\  -.  n  =  0
)  ->  ( n  -  1 )  e. 
NN0 )
17 eqidd 2383 . . . . . . . . . 10  |-  ( m  =  ( n  - 
1 )  ->  0  =  0 )
18 fconstmpt 4957 . . . . . . . . . . 11  |-  ( NN0 
X.  { 0 } )  =  ( m  e.  NN0  |->  0 )
199, 18eqtri 2411 . . . . . . . . . 10  |-  (coeff ` 
0p )  =  ( m  e.  NN0  |->  0 )
20 c0ex 9501 . . . . . . . . . 10  |-  0  e.  _V
2117, 19, 20fvmpt 5857 . . . . . . . . 9  |-  ( ( n  -  1 )  e.  NN0  ->  ( (coeff `  0p ) `
 ( n  - 
1 ) )  =  0 )
2216, 21syl 16 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  -.  n  =  0
)  ->  ( (coeff `  0p ) `  ( n  -  1
) )  =  0 )
2310, 22ifeqda 3890 . . . . . . 7  |-  ( n  e.  NN0  ->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) )  =  0 )
2423mpteq2ia 4449 . . . . . 6  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) ) )  =  ( n  e.  NN0  |->  0 )
258, 9, 243eqtr4ri 2422 . . . . 5  |-  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) ) )  =  (coeff ` 
0p )
267, 25eqtr4i 2414 . . . 4  |-  (coeff `  ( 0p  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  0p ) `
 ( n  - 
1 ) ) ) )
27 oveq1 6203 . . . . 5  |-  ( F  =  0p  -> 
( F  oF  x.  Xp )  =  ( 0p  oF  x.  Xp ) )
2827fveq2d 5778 . . . 4  |-  ( F  =  0p  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  (coeff `  ( 0p  oF  x.  Xp ) ) )
29 simpl 455 . . . . . . . 8  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  F  =  0p )
3029fveq2d 5778 . . . . . . 7  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  (coeff `  F
)  =  (coeff ` 
0p ) )
3130fveq1d 5776 . . . . . 6  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  ( (coeff `  F ) `  (
n  -  1 ) )  =  ( (coeff `  0p ) `
 ( n  - 
1 ) ) )
3231ifeq2d 3876 . . . . 5  |-  ( ( F  =  0p  /\  n  e.  NN0 )  ->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  (
n  -  1 ) ) )  =  if ( n  =  0 ,  0 ,  ( (coeff `  0p
) `  ( n  -  1 ) ) ) )
3332mpteq2dva 4453 . . . 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 2449 . . 3  |-  ( F  =  0p  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
3534adantl 464 . 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 455 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  F  e.  (Poly `  RR ) )
37 elsncg 3967 . . . . . 6  |-  ( F  e.  (Poly `  RR )  ->  ( F  e. 
{ 0p }  <->  F  =  0p ) )
3837notbid 292 . . . . 5  |-  ( F  e.  (Poly `  RR )  ->  ( -.  F  e.  { 0p }  <->  -.  F  =  0p ) )
3938biimpar 483 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  -.  F  e.  { 0p } )
4036, 39eldifd 3400 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  -.  F  =  0p )  ->  F  e.  ( (Poly `  RR )  \  { 0p }
) )
41 plymulx0 28687 . . 3  |-  ( F  e.  ( (Poly `  RR )  \  { 0p } )  -> 
(coeff `  ( F  oF  x.  Xp ) )  =  ( n  e.  NN0  |->  if ( n  =  0 ,  0 ,  ( (coeff `  F ) `  ( n  -  1 ) ) ) ) )
4240, 41syl 16 . 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 789 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 367    = wceq 1399    e. wcel 1826    =/= wne 2577    \ cdif 3386    C_ wss 3389   ifcif 3857   {csn 3944    |-> cmpt 4425    X. cxp 4911   ` cfv 5496  (class class class)co 6196    oFcof 6437   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    x. cmul 9408    - cmin 9718   NNcn 10452   NN0cn0 10712   0pc0p 22161  Polycply 22666   Xpcidp 22667  coeffccoe 22668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-rlim 13314  df-sum 13511  df-0p 22162  df-ply 22670  df-idp 22671  df-coe 22672  df-dgr 22673
This theorem is referenced by: (None)
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