Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bpolydif Structured version   Unicode version

Theorem bpolydif 26101
Description: Calculate the difference between successive values of the Bernoulli polynomials. (Contributed by Scott Fenton, 16-May-2014.) (Proof shortened by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
bpolydif  |-  ( ( N  e.  NN  /\  X  e.  CC )  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  - 
1 ) ) ) )

Proof of Theorem bpolydif
Dummy variables  k  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6088 . . . . . 6  |-  ( n  =  k  ->  (
n BernPoly  ( X  +  1 ) )  =  ( k BernPoly  ( X  + 
1 ) ) )
2 oveq1 6088 . . . . . 6  |-  ( n  =  k  ->  (
n BernPoly  X )  =  ( k BernPoly  X ) )
31, 2oveq12d 6099 . . . . 5  |-  ( n  =  k  ->  (
( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X
) )  =  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) ) )
4 id 20 . . . . . 6  |-  ( n  =  k  ->  n  =  k )
5 oveq1 6088 . . . . . . 7  |-  ( n  =  k  ->  (
n  -  1 )  =  ( k  - 
1 ) )
65oveq2d 6097 . . . . . 6  |-  ( n  =  k  ->  ( X ^ ( n  - 
1 ) )  =  ( X ^ (
k  -  1 ) ) )
74, 6oveq12d 6099 . . . . 5  |-  ( n  =  k  ->  (
n  x.  ( X ^ ( n  - 
1 ) ) )  =  ( k  x.  ( X ^ (
k  -  1 ) ) ) )
83, 7eqeq12d 2450 . . . 4  |-  ( n  =  k  ->  (
( ( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X ) )  =  ( n  x.  ( X ^ ( n  - 
1 ) ) )  <-> 
( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) ) )
98imbi2d 308 . . 3  |-  ( n  =  k  ->  (
( X  e.  CC  ->  ( ( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X ) )  =  ( n  x.  ( X ^ ( n  - 
1 ) ) ) )  <->  ( X  e.  CC  ->  ( (
k BernPoly  ( X  +  1 ) )  -  (
k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) ) ) )
10 oveq1 6088 . . . . . 6  |-  ( n  =  N  ->  (
n BernPoly  ( X  +  1 ) )  =  ( N BernPoly  ( X  + 
1 ) ) )
11 oveq1 6088 . . . . . 6  |-  ( n  =  N  ->  (
n BernPoly  X )  =  ( N BernPoly  X ) )
1210, 11oveq12d 6099 . . . . 5  |-  ( n  =  N  ->  (
( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X
) )  =  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) ) )
13 id 20 . . . . . 6  |-  ( n  =  N  ->  n  =  N )
14 oveq1 6088 . . . . . . 7  |-  ( n  =  N  ->  (
n  -  1 )  =  ( N  - 
1 ) )
1514oveq2d 6097 . . . . . 6  |-  ( n  =  N  ->  ( X ^ ( n  - 
1 ) )  =  ( X ^ ( N  -  1 ) ) )
1613, 15oveq12d 6099 . . . . 5  |-  ( n  =  N  ->  (
n  x.  ( X ^ ( n  - 
1 ) ) )  =  ( N  x.  ( X ^ ( N  -  1 ) ) ) )
1712, 16eqeq12d 2450 . . . 4  |-  ( n  =  N  ->  (
( ( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X ) )  =  ( n  x.  ( X ^ ( n  - 
1 ) ) )  <-> 
( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  - 
1 ) ) ) ) )
1817imbi2d 308 . . 3  |-  ( n  =  N  ->  (
( X  e.  CC  ->  ( ( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X ) )  =  ( n  x.  ( X ^ ( n  - 
1 ) ) ) )  <->  ( X  e.  CC  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  - 
1 ) ) ) ) ) )
19 simp1 957 . . . . 5  |-  ( ( n  e.  NN  /\  A. k  e.  ( 1 ... ( n  - 
1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  X  e.  CC )  ->  n  e.  NN )
20 simp3 959 . . . . 5  |-  ( ( n  e.  NN  /\  A. k  e.  ( 1 ... ( n  - 
1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  X  e.  CC )  ->  X  e.  CC )
21 simpl3 962 . . . . . 6  |-  ( ( ( n  e.  NN  /\ 
A. k  e.  ( 1 ... ( n  -  1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  X  e.  CC )  /\  m  e.  ( 1 ... (
n  -  1 ) ) )  ->  X  e.  CC )
22 oveq1 6088 . . . . . . . . . . 11  |-  ( k  =  m  ->  (
k BernPoly  ( X  +  1 ) )  =  ( m BernPoly  ( X  + 
1 ) ) )
23 oveq1 6088 . . . . . . . . . . 11  |-  ( k  =  m  ->  (
k BernPoly  X )  =  ( m BernPoly  X ) )
2422, 23oveq12d 6099 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) )  =  ( ( m BernPoly  ( X  +  1 ) )  -  ( m BernPoly  X
) ) )
25 id 20 . . . . . . . . . . 11  |-  ( k  =  m  ->  k  =  m )
26 oveq1 6088 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
k  -  1 )  =  ( m  - 
1 ) )
2726oveq2d 6097 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( X ^ ( k  - 
1 ) )  =  ( X ^ (
m  -  1 ) ) )
2825, 27oveq12d 6099 . . . . . . . . . 10  |-  ( k  =  m  ->  (
k  x.  ( X ^ ( k  - 
1 ) ) )  =  ( m  x.  ( X ^ (
m  -  1 ) ) ) )
2924, 28eqeq12d 2450 . . . . . . . . 9  |-  ( k  =  m  ->  (
( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) )  <-> 
( ( m BernPoly  ( X  +  1 ) )  -  ( m BernPoly  X ) )  =  ( m  x.  ( X ^ ( m  - 
1 ) ) ) ) )
3029imbi2d 308 . . . . . . . 8  |-  ( k  =  m  ->  (
( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  <->  ( X  e.  CC  ->  ( (
m BernPoly  ( X  +  1 ) )  -  (
m BernPoly  X ) )  =  ( m  x.  ( X ^ ( m  - 
1 ) ) ) ) ) )
3130rspccva 3051 . . . . . . 7  |-  ( ( A. k  e.  ( 1 ... ( n  -  1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  m  e.  ( 1 ... (
n  -  1 ) ) )  ->  ( X  e.  CC  ->  ( ( m BernPoly  ( X  +  1 ) )  -  ( m BernPoly  X
) )  =  ( m  x.  ( X ^ ( m  - 
1 ) ) ) ) )
32313ad2antl2 1120 . . . . . 6  |-  ( ( ( n  e.  NN  /\ 
A. k  e.  ( 1 ... ( n  -  1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  X  e.  CC )  /\  m  e.  ( 1 ... (
n  -  1 ) ) )  ->  ( X  e.  CC  ->  ( ( m BernPoly  ( X  +  1 ) )  -  ( m BernPoly  X
) )  =  ( m  x.  ( X ^ ( m  - 
1 ) ) ) ) )
3321, 32mpd 15 . . . . 5  |-  ( ( ( n  e.  NN  /\ 
A. k  e.  ( 1 ... ( n  -  1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X ) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  X  e.  CC )  /\  m  e.  ( 1 ... (
n  -  1 ) ) )  ->  (
( m BernPoly  ( X  +  1 ) )  -  ( m BernPoly  X
) )  =  ( m  x.  ( X ^ ( m  - 
1 ) ) ) )
3419, 20, 33bpolydiflem 26100 . . . 4  |-  ( ( n  e.  NN  /\  A. k  e.  ( 1 ... ( n  - 
1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  /\  X  e.  CC )  ->  (
( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X
) )  =  ( n  x.  ( X ^ ( n  - 
1 ) ) ) )
35343exp 1152 . . 3  |-  ( n  e.  NN  ->  ( A. k  e.  (
1 ... ( n  - 
1 ) ) ( X  e.  CC  ->  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) )  =  ( k  x.  ( X ^ ( k  - 
1 ) ) ) )  ->  ( X  e.  CC  ->  ( (
n BernPoly  ( X  +  1 ) )  -  (
n BernPoly  X ) )  =  ( n  x.  ( X ^ ( n  - 
1 ) ) ) ) ) )
369, 18, 35nnsinds 25492 . 2  |-  ( N  e.  NN  ->  ( X  e.  CC  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^
( N  -  1 ) ) ) ) )
3736imp 419 1  |-  ( ( N  e.  NN  /\  X  e.  CC )  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^ ( N  - 
1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   ...cfz 11043   ^cexp 11382   BernPoly cbp 26092
This theorem is referenced by:  fsumkthpow  26102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-pred 25439  df-wrecs 25531  df-bpoly 26093
  Copyright terms: Public domain W3C validator