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Theorem gsummptshft 16829
Description: Index shift of a finite group sum over a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.)
Hypotheses
Ref Expression
gsummptshft.b  |-  B  =  ( Base `  G
)
gsummptshft.z  |-  .0.  =  ( 0g `  G )
gsummptshft.g  |-  ( ph  ->  G  e. CMnd )
gsummptshft.k  |-  ( ph  ->  K  e.  ZZ )
gsummptshft.m  |-  ( ph  ->  M  e.  ZZ )
gsummptshft.n  |-  ( ph  ->  N  e.  ZZ )
gsummptshft.a  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  B )
gsummptshft.c  |-  ( j  =  ( k  -  K )  ->  A  =  C )
Assertion
Ref Expression
gsummptshft  |-  ( ph  ->  ( G  gsumg  ( j  e.  ( M ... N ) 
|->  A ) )  =  ( G  gsumg  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  C ) ) )
Distinct variable groups:    A, k    B, j    C, j    j, k, K    j, M, k   
j, N, k    ph, j,
k
Allowed substitution hints:    A( j)    B( k)    C( k)    G( j, k)    .0. ( j, k)

Proof of Theorem gsummptshft
StepHypRef Expression
1 gsummptshft.b . . 3  |-  B  =  ( Base `  G
)
2 gsummptshft.z . . 3  |-  .0.  =  ( 0g `  G )
3 gsummptshft.g . . 3  |-  ( ph  ->  G  e. CMnd )
4 ovex 6320 . . . 4  |-  ( M ... N )  e. 
_V
54a1i 11 . . 3  |-  ( ph  ->  ( M ... N
)  e.  _V )
6 gsummptshft.a . . . 4  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  B )
7 eqid 2467 . . . 4  |-  ( j  e.  ( M ... N )  |->  A )  =  ( j  e.  ( M ... N
)  |->  A )
86, 7fmptd 6056 . . 3  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A ) : ( M ... N ) --> B )
9 fzfid 12063 . . . 4  |-  ( ph  ->  ( M ... N
)  e.  Fin )
10 fvex 5882 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
112, 10eqeltri 2551 . . . . 5  |-  .0.  e.  _V
1211a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
137, 9, 6, 12fsuppmptdm 7852 . . 3  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A ) finSupp  .0.  )
14 gsummptshft.k . . . 4  |-  ( ph  ->  K  e.  ZZ )
15 gsummptshft.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
16 gsummptshft.n . . . 4  |-  ( ph  ->  N  e.  ZZ )
1714, 15, 16mptfzshft 13573 . . 3  |-  ( ph  ->  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) : ( ( M  +  K ) ... ( N  +  K ) ) -1-1-onto-> ( M ... N ) )
181, 2, 3, 5, 8, 13, 17gsumf1o 16797 . 2  |-  ( ph  ->  ( G  gsumg  ( j  e.  ( M ... N ) 
|->  A ) )  =  ( G  gsumg  ( ( j  e.  ( M ... N
)  |->  A )  o.  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) ) ) )
19 elfzelz 11700 . . . . . . . . . 10  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  k  e.  ZZ )
2019zcnd 10979 . . . . . . . . 9  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  k  e.  CC )
2114zcnd 10979 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
22 npcan 9841 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( ( k  -  K )  +  K
)  =  k )
2320, 21, 22syl2anr 478 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( k  -  K
)  +  K )  =  k )
24 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )
2523, 24eqeltrd 2555 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( k  -  K
)  +  K )  e.  ( ( M  +  K ) ... ( N  +  K
) ) )
2615, 16jca 532 . . . . . . . . 9  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2726adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2819adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  k  e.  ZZ )
2914adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  K  e.  ZZ )
3028, 29zsubcld 10983 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  ZZ )
31 fzaddel 11730 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( ( k  -  K )  e.  ZZ  /\  K  e.  ZZ ) )  -> 
( ( k  -  K )  e.  ( M ... N )  <-> 
( ( k  -  K )  +  K
)  e.  ( ( M  +  K ) ... ( N  +  K ) ) ) )
3227, 30, 29, 31syl12anc 1226 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( k  -  K
)  e.  ( M ... N )  <->  ( (
k  -  K )  +  K )  e.  ( ( M  +  K ) ... ( N  +  K )
) ) )
3325, 32mpbird 232 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  ( M ... N ) )
3433ralrimiva 2881 . . . . 5  |-  ( ph  ->  A. k  e.  ( ( M  +  K
) ... ( N  +  K ) ) ( k  -  K )  e.  ( M ... N ) )
35 eqidd 2468 . . . . 5  |-  ( ph  ->  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( k  -  K ) ) )
36 eqidd 2468 . . . . 5  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A )  =  ( j  e.  ( M ... N )  |->  A ) )
3734, 35, 36fmptcos 6067 . . . 4  |-  ( ph  ->  ( ( j  e.  ( M ... N
)  |->  A )  o.  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  [_ ( k  -  K
)  /  j ]_ A ) )
38 nfv 1683 . . . . . . 7  |-  F/ j
ph
39 nfv 1683 . . . . . . 7  |-  F/ j  k  e.  ( ( M  +  K ) ... ( N  +  K ) )
4038, 39nfan 1875 . . . . . 6  |-  F/ j ( ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )
41 nfcv 2629 . . . . . . 7  |-  F/_ j C
4241a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  F/_ j C )
43 ovex 6320 . . . . . . 7  |-  ( k  -  K )  e. 
_V
4443a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  _V )
45 gsummptshft.c . . . . . . 7  |-  ( j  =  ( k  -  K )  ->  A  =  C )
4645adantl 466 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  /\  j  =  ( k  -  K ) )  ->  A  =  C )
4740, 42, 44, 46csbiedf 3461 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  [_ (
k  -  K )  /  j ]_ A  =  C )
4847mpteq2dva 4539 . . . 4  |-  ( ph  ->  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  [_ ( k  -  K
)  /  j ]_ A )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  C ) )
4937, 48eqtrd 2508 . . 3  |-  ( ph  ->  ( ( j  e.  ( M ... N
)  |->  A )  o.  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  C ) )
5049oveq2d 6311 . 2  |-  ( ph  ->  ( G  gsumg  ( ( j  e.  ( M ... N
)  |->  A )  o.  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) ) )  =  ( G  gsumg  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  C ) ) )
5118, 50eqtrd 2508 1  |-  ( ph  ->  ( G  gsumg  ( j  e.  ( M ... N ) 
|->  A ) )  =  ( G  gsumg  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   _Vcvv 3118   [_csb 3440    |-> cmpt 4511    o. ccom 5009   ` cfv 5594  (class class class)co 6295   CCcc 9502    + caddc 9507    - cmin 9817   ZZcz 10876   ...cfz 11684   Basecbs 14507   0gc0g 14712    gsumg cgsu 14713  CMndccmn 16671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-0g 14714  df-gsum 14715  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-cntz 16227  df-cmn 16673
This theorem is referenced by:  srgbinomlem4  17066  cpmadugsumlemF  19246
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