MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coemul Structured version   Unicode version

Theorem coemul 20172
Description: A coefficient of a product of polynomials. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
coefv0.1  |-  A  =  (coeff `  F )
coeadd.2  |-  B  =  (coeff `  G )
Assertion
Ref Expression
coemul  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  N )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( B `
 ( N  -  k ) ) ) )
Distinct variable groups:    A, k    B, k    k, F    k, G    k, N    S, k

Proof of Theorem coemul
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 coefv0.1 . . . . . 6  |-  A  =  (coeff `  F )
2 coeadd.2 . . . . . 6  |-  B  =  (coeff `  G )
3 eqid 2438 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
4 eqid 2438 . . . . . 6  |-  (deg `  G )  =  (deg
`  G )
51, 2, 3, 4coemullem 20170 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) )  =  ( n  e. 
NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A `  k )  x.  ( B `  ( n  -  k ) ) ) )  /\  (deg `  ( F  o F  x.  G ) )  <_  ( (deg `  F )  +  (deg
`  G ) ) ) )
65simpld 447 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  x.  G
) )  =  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( B `  (
n  -  k ) ) ) ) )
76fveq1d 5732 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 N )  =  ( ( n  e. 
NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A `  k )  x.  ( B `  ( n  -  k ) ) ) ) `  N
) )
8 oveq2 6091 . . . . 5  |-  ( n  =  N  ->  (
0 ... n )  =  ( 0 ... N
) )
9 oveq1 6090 . . . . . . . 8  |-  ( n  =  N  ->  (
n  -  k )  =  ( N  -  k ) )
109fveq2d 5734 . . . . . . 7  |-  ( n  =  N  ->  ( B `  ( n  -  k ) )  =  ( B `  ( N  -  k
) ) )
1110oveq2d 6099 . . . . . 6  |-  ( n  =  N  ->  (
( A `  k
)  x.  ( B `
 ( n  -  k ) ) )  =  ( ( A `
 k )  x.  ( B `  ( N  -  k )
) ) )
1211adantr 453 . . . . 5  |-  ( ( n  =  N  /\  k  e.  ( 0 ... n ) )  ->  ( ( A `
 k )  x.  ( B `  (
n  -  k ) ) )  =  ( ( A `  k
)  x.  ( B `
 ( N  -  k ) ) ) )
138, 12sumeq12dv 12502 . . . 4  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( B `  (
n  -  k ) ) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( B `
 ( N  -  k ) ) ) )
14 eqid 2438 . . . 4  |-  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( B `  (
n  -  k ) ) ) )  =  ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n ) ( ( A `  k
)  x.  ( B `
 ( n  -  k ) ) ) )
15 sumex 12483 . . . 4  |-  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( B `  ( N  -  k )
) )  e.  _V
1613, 14, 15fvmpt 5808 . . 3  |-  ( N  e.  NN0  ->  ( ( n  e.  NN0  |->  sum_ k  e.  ( 0 ... n
) ( ( A `
 k )  x.  ( B `  (
n  -  k ) ) ) ) `  N )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( B `
 ( N  -  k ) ) ) )
177, 16sylan9eq 2490 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  N  e.  NN0 )  ->  ( (coeff `  ( F  o F  x.  G ) ) `
 N )  = 
sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  ( B `  ( N  -  k ) ) ) )
18173impa 1149 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( (coeff `  ( F  o F  x.  G ) ) `  N )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( B `
 ( N  -  k ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083    o Fcof 6305   0cc0 8992    + caddc 8995    x. cmul 8997    <_ cle 9123    - cmin 9293   NN0cn0 10223   ...cfz 11045   sum_csu 12481  Polycply 20105  coeffccoe 20107  degcdgr 20108
This theorem is referenced by:  coemulhi  20174  coemulc  20175  vieta1lem2  20230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-0p 19564  df-ply 20109  df-coe 20111  df-dgr 20112
  Copyright terms: Public domain W3C validator