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Theorem bpolydif 24790
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 5865 . . . . . 6  |-  ( n  =  k  ->  (
n BernPoly  ( X  +  1 ) )  =  ( k BernPoly  ( X  + 
1 ) ) )
2 oveq1 5865 . . . . . 6  |-  ( n  =  k  ->  (
n BernPoly  X )  =  ( k BernPoly  X ) )
31, 2oveq12d 5876 . . . . 5  |-  ( n  =  k  ->  (
( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X
) )  =  ( ( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) ) )
4 id 19 . . . . . 6  |-  ( n  =  k  ->  n  =  k )
5 oveq1 5865 . . . . . . 7  |-  ( n  =  k  ->  (
n  -  1 )  =  ( k  - 
1 ) )
65oveq2d 5874 . . . . . 6  |-  ( n  =  k  ->  ( X ^ ( n  - 
1 ) )  =  ( X ^ (
k  -  1 ) ) )
74, 6oveq12d 5876 . . . . 5  |-  ( n  =  k  ->  (
n  x.  ( X ^ ( n  - 
1 ) ) )  =  ( k  x.  ( X ^ (
k  -  1 ) ) ) )
83, 7eqeq12d 2297 . . . 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 307 . . 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 5865 . . . . . 6  |-  ( n  =  N  ->  (
n BernPoly  ( X  +  1 ) )  =  ( N BernPoly  ( X  + 
1 ) ) )
11 oveq1 5865 . . . . . 6  |-  ( n  =  N  ->  (
n BernPoly  X )  =  ( N BernPoly  X ) )
1210, 11oveq12d 5876 . . . . 5  |-  ( n  =  N  ->  (
( n BernPoly  ( X  +  1 ) )  -  ( n BernPoly  X
) )  =  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) ) )
13 id 19 . . . . . 6  |-  ( n  =  N  ->  n  =  N )
14 oveq1 5865 . . . . . . 7  |-  ( n  =  N  ->  (
n  -  1 )  =  ( N  - 
1 ) )
1514oveq2d 5874 . . . . . 6  |-  ( n  =  N  ->  ( X ^ ( n  - 
1 ) )  =  ( X ^ ( N  -  1 ) ) )
1613, 15oveq12d 5876 . . . . 5  |-  ( n  =  N  ->  (
n  x.  ( X ^ ( n  - 
1 ) ) )  =  ( N  x.  ( X ^ ( N  -  1 ) ) ) )
1712, 16eqeq12d 2297 . . . 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 307 . . 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 955 . . . . 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 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 )  ->  X  e.  CC )
21 simpl3 960 . . . . . 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 5865 . . . . . . . . . . 11  |-  ( k  =  m  ->  (
k BernPoly  ( X  +  1 ) )  =  ( m BernPoly  ( X  + 
1 ) ) )
23 oveq1 5865 . . . . . . . . . . 11  |-  ( k  =  m  ->  (
k BernPoly  X )  =  ( m BernPoly  X ) )
2422, 23oveq12d 5876 . . . . . . . . . 10  |-  ( k  =  m  ->  (
( k BernPoly  ( X  +  1 ) )  -  ( k BernPoly  X
) )  =  ( ( m BernPoly  ( X  +  1 ) )  -  ( m BernPoly  X
) ) )
25 id 19 . . . . . . . . . . 11  |-  ( k  =  m  ->  k  =  m )
26 oveq1 5865 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
k  -  1 )  =  ( m  - 
1 ) )
2726oveq2d 5874 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( X ^ ( k  - 
1 ) )  =  ( X ^ (
m  -  1 ) ) )
2825, 27oveq12d 5876 . . . . . . . . . 10  |-  ( k  =  m  ->  (
k  x.  ( X ^ ( k  - 
1 ) ) )  =  ( m  x.  ( X ^ (
m  -  1 ) ) ) )
2924, 28eqeq12d 2297 . . . . . . . . 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 307 . . . . . . . 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 2883 . . . . . . 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 1118 . . . . . 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 14 . . . . 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 24789 . . . 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 1150 . . 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 24217 . 2  |-  ( N  e.  NN  ->  ( X  e.  CC  ->  ( ( N BernPoly  ( X  +  1 ) )  -  ( N BernPoly  X ) )  =  ( N  x.  ( X ^
( N  -  1 ) ) ) ) )
3736imp 418 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   ...cfz 10782   ^cexp 11104   BernPoly cbp 24781
This theorem is referenced by:  fsumkthpow  24791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-pred 24168  df-bpoly 24782
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