Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsumshftd Structured version   Unicode version

Theorem fsumshftd 35134
Description: Index shift of a finite sum with a weaker "implicit substitution" hypothesis than fsumshft 13620. The proof demonstrates how this can be derived starting from from fsumshft 13620. (Contributed by NM, 1-Nov-2019.)
Hypotheses
Ref Expression
fsumshftd.1  |-  ( ph  ->  K  e.  ZZ )
fsumshftd.2  |-  ( ph  ->  M  e.  ZZ )
fsumshftd.3  |-  ( ph  ->  N  e.  ZZ )
fsumshftd.4  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
fsumshftd.5  |-  ( (
ph  /\  j  =  ( k  -  K
) )  ->  A  =  B )
Assertion
Ref Expression
fsumshftd  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) B )
Distinct variable groups:    A, k    B, j    j, k, K   
j, M, k    j, N, k    ph, j, k
Allowed substitution hints:    A( j)    B( k)

Proof of Theorem fsumshftd
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfcv 2558 . . 3  |-  F/_ w A
2 nfcsb1v 3381 . . 3  |-  F/_ j [_ w  /  j ]_ A
3 csbeq1a 3374 . . 3  |-  ( j  =  w  ->  A  =  [_ w  /  j ]_ A )
41, 2, 3cbvsumi 13544 . 2  |-  sum_ j  e.  ( M ... N
) A  =  sum_ w  e.  ( M ... N ) [_ w  /  j ]_ A
5 fsumshftd.1 . . . 4  |-  ( ph  ->  K  e.  ZZ )
6 fsumshftd.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
7 fsumshftd.3 . . . 4  |-  ( ph  ->  N  e.  ZZ )
8 nfv 1722 . . . . . 6  |-  F/ j ( ph  /\  w  e.  ( M ... N
) )
92nfel1 2574 . . . . . 6  |-  F/ j
[_ w  /  j ]_ A  e.  CC
108, 9nfim 1942 . . . . 5  |-  F/ j ( ( ph  /\  w  e.  ( M ... N ) )  ->  [_ w  /  j ]_ A  e.  CC )
11 eleq1 2468 . . . . . . 7  |-  ( j  =  w  ->  (
j  e.  ( M ... N )  <->  w  e.  ( M ... N ) ) )
1211anbi2d 701 . . . . . 6  |-  ( j  =  w  ->  (
( ph  /\  j  e.  ( M ... N
) )  <->  ( ph  /\  w  e.  ( M ... N ) ) ) )
133eleq1d 2465 . . . . . 6  |-  ( j  =  w  ->  ( A  e.  CC  <->  [_ w  / 
j ]_ A  e.  CC ) )
1412, 13imbi12d 318 . . . . 5  |-  ( j  =  w  ->  (
( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )  <->  ( ( ph  /\  w  e.  ( M ... N
) )  ->  [_ w  /  j ]_ A  e.  CC ) ) )
15 fsumshftd.4 . . . . 5  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
1610, 14, 15chvar 2034 . . . 4  |-  ( (
ph  /\  w  e.  ( M ... N ) )  ->  [_ w  / 
j ]_ A  e.  CC )
17 csbeq1 3368 . . . 4  |-  ( w  =  ( k  -  K )  ->  [_ w  /  j ]_ A  =  [_ ( k  -  K )  /  j ]_ A )
185, 6, 7, 16, 17fsumshft 13620 . . 3  |-  ( ph  -> 
sum_ w  e.  ( M ... N ) [_ w  /  j ]_ A  =  sum_ k  e.  ( ( M  +  K
) ... ( N  +  K ) ) [_ ( k  -  K
)  /  j ]_ A )
19 ovex 6246 . . . . . 6  |-  ( k  -  K )  e. 
_V
2019a1i 11 . . . . 5  |-  ( ph  ->  ( k  -  K
)  e.  _V )
21 fsumshftd.5 . . . . 5  |-  ( (
ph  /\  j  =  ( k  -  K
) )  ->  A  =  B )
2220, 21csbied 3392 . . . 4  |-  ( ph  ->  [_ ( k  -  K )  /  j ]_ A  =  B
)
2322sumeq2sdv 13551 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ( M  +  K
) ... ( N  +  K ) ) [_ ( k  -  K
)  /  j ]_ A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) B )
2418, 23eqtrd 2437 . 2  |-  ( ph  -> 
sum_ w  e.  ( M ... N ) [_ w  /  j ]_ A  =  sum_ k  e.  ( ( M  +  K
) ... ( N  +  K ) ) B )
254, 24syl5eq 2449 1  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   _Vcvv 3051   [_csb 3365  (class class class)co 6218   CCcc 9423    + caddc 9428    - cmin 9740   ZZcz 10803   ...cfz 11615   sum_csu 13533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-inf2 7994  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
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 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-1o 7070  df-oadd 7074  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-sup 7838  df-oi 7872  df-card 8255  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-n0 10735  df-z 10804  df-uz 11024  df-rp 11162  df-fz 11616  df-fzo 11740  df-seq 12034  df-exp 12093  df-hash 12331  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094  df-clim 13336  df-sum 13534
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator