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Theorem fsumshftdOLD 31940
Description: Obsolete version of fsumshftd 31939 as of 1-Nov-2019. (Contributed by NM, 1-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
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
fsumshftdOLD  |-  ( 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 fsumshftdOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsumshftd.1 . . 3  |-  ( ph  ->  K  e.  ZZ )
2 fsumshftd.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 fsumshftd.3 . . 3  |-  ( ph  ->  N  e.  ZZ )
4 nfv 1726 . . . . 5  |-  F/ j ( ph  /\  x  e.  ( M ... N
) )
5 nfcsb1v 3386 . . . . . 6  |-  F/_ j [_ x  /  j ]_ A
65nfel1 2578 . . . . 5  |-  F/ j
[_ x  /  j ]_ A  e.  CC
74, 6nfim 1946 . . . 4  |-  F/ j ( ( ph  /\  x  e.  ( M ... N ) )  ->  [_ x  /  j ]_ A  e.  CC )
8 eleq1 2472 . . . . . 6  |-  ( j  =  x  ->  (
j  e.  ( M ... N )  <->  x  e.  ( M ... N ) ) )
98anbi2d 702 . . . . 5  |-  ( j  =  x  ->  (
( ph  /\  j  e.  ( M ... N
) )  <->  ( ph  /\  x  e.  ( M ... N ) ) ) )
10 csbeq1a 3379 . . . . . 6  |-  ( j  =  x  ->  A  =  [_ x  /  j ]_ A )
1110eleq1d 2469 . . . . 5  |-  ( j  =  x  ->  ( A  e.  CC  <->  [_ x  / 
j ]_ A  e.  CC ) )
129, 11imbi12d 318 . . . 4  |-  ( j  =  x  ->  (
( ( ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )  <->  ( ( ph  /\  x  e.  ( M ... N
) )  ->  [_ x  /  j ]_ A  e.  CC ) ) )
13 fsumshftd.4 . . . 4  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
147, 12, 13chvar 2038 . . 3  |-  ( (
ph  /\  x  e.  ( M ... N ) )  ->  [_ x  / 
j ]_ A  e.  CC )
15 csbeq1 3373 . . 3  |-  ( x  =  ( y  -  K )  ->  [_ x  /  j ]_ A  =  [_ ( y  -  K )  /  j ]_ A )
161, 2, 3, 14, 15fsumshft 13651 . 2  |-  ( ph  -> 
sum_ x  e.  ( M ... N ) [_ x  /  j ]_ A  =  sum_ y  e.  ( ( M  +  K
) ... ( N  +  K ) ) [_ ( y  -  K
)  /  j ]_ A )
17 nfcv 2562 . . . . 5  |-  F/_ x A
1817, 5, 10cbvsumi 13573 . . . 4  |-  sum_ j  e.  ( M ... N
) A  =  sum_ x  e.  ( M ... N ) [_ x  /  j ]_ A
1918eqcomi 2413 . . 3  |-  sum_ x  e.  ( M ... N
) [_ x  /  j ]_ A  =  sum_ j  e.  ( M ... N ) A
2019a1i 11 . 2  |-  ( ph  -> 
sum_ x  e.  ( M ... N ) [_ x  /  j ]_ A  =  sum_ j  e.  ( M ... N ) A )
21 nfcv 2562 . . . 4  |-  F/_ y [_ ( k  -  K
)  /  j ]_ A
22 nfcv 2562 . . . 4  |-  F/_ k [_ ( y  -  K
)  /  j ]_ A
23 oveq1 6239 . . . . 5  |-  ( k  =  y  ->  (
k  -  K )  =  ( y  -  K ) )
2423csbeq1d 3377 . . . 4  |-  ( k  =  y  ->  [_ (
k  -  K )  /  j ]_ A  =  [_ ( y  -  K )  /  j ]_ A )
2521, 22, 24cbvsumi 13573 . . 3  |-  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) [_ ( k  -  K )  /  j ]_ A  =  sum_ y  e.  ( ( M  +  K ) ... ( N  +  K
) ) [_ (
y  -  K )  /  j ]_ A
26 ovex 6260 . . . . . 6  |-  ( k  -  K )  e. 
_V
2726a1i 11 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  _V )
28 fsumshftd.5 . . . . . 6  |-  ( (
ph  /\  j  =  ( k  -  K
) )  ->  A  =  B )
2928adantlr 713 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  /\  j  =  ( k  -  K ) )  ->  A  =  B )
3027, 29csbied 3397 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  [_ (
k  -  K )  /  j ]_ A  =  B )
3130sumeq2dv 13579 . . 3  |-  ( ph  -> 
sum_ k  e.  ( ( M  +  K
) ... ( N  +  K ) ) [_ ( k  -  K
)  /  j ]_ A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) B )
3225, 31syl5eqr 2455 . 2  |-  ( ph  -> 
sum_ y  e.  ( ( M  +  K
) ... ( N  +  K ) ) [_ ( y  -  K
)  /  j ]_ A  =  sum_ k  e.  ( ( M  +  K ) ... ( N  +  K )
) B )
3316, 20, 323eqtr3d 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 1403    e. wcel 1840   _Vcvv 3056   [_csb 3370  (class class class)co 6232   CCcc 9438    + caddc 9443    - cmin 9759   ZZcz 10823   ...cfz 11641   sum_csu 13562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-oi 7887  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-fz 11642  df-fzo 11766  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-sum 13563
This theorem is referenced by: (None)
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