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Theorem gsummptshft 16429
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 6116 . . . 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 2443 . . . 4  |-  ( j  e.  ( M ... N )  |->  A )  =  ( j  e.  ( M ... N
)  |->  A )
86, 7fmptd 5867 . . 3  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A ) : ( M ... N ) --> B )
9 fzfid 11795 . . . 4  |-  ( ph  ->  ( M ... N
)  e.  Fin )
10 fvex 5701 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
112, 10eqeltri 2513 . . . . 5  |-  .0.  e.  _V
1211a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
137, 9, 6, 12fsuppmptdm 7631 . . 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 13245 . . 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 16398 . 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 11453 . . . . . . . . . 10  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  k  e.  ZZ )
2019zcnd 10748 . . . . . . . . 9  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  k  e.  CC )
2114zcnd 10748 . . . . . . . . 9  |-  ( ph  ->  K  e.  CC )
22 npcan 9619 . . . . . . . . 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 2517 . . . . . . 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 10752 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  ZZ )
31 fzaddel 11493 . . . . . . . 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 1216 . . . . . . 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 2799 . . . . 5  |-  ( ph  ->  A. k  e.  ( ( M  +  K
) ... ( N  +  K ) ) ( k  -  K )  e.  ( M ... N ) )
35 eqidd 2444 . . . . 5  |-  ( ph  ->  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( k  -  K ) ) )
36 eqidd 2444 . . . . 5  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A )  =  ( j  e.  ( M ... N )  |->  A ) )
3734, 35, 36fmptcos 5878 . . . 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 1673 . . . . . . 7  |-  F/ j
ph
39 nfv 1673 . . . . . . 7  |-  F/ j  k  e.  ( ( M  +  K ) ... ( N  +  K ) )
4038, 39nfan 1861 . . . . . 6  |-  F/ j ( ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )
41 nfcv 2579 . . . . . . 7  |-  F/_ j C
4241a1i 11 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  F/_ j C )
43 ovex 6116 . . . . . . 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 3309 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  [_ (
k  -  K )  /  j ]_ A  =  C )
4847mpteq2dva 4378 . . . 4  |-  ( ph  ->  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  [_ ( k  -  K
)  /  j ]_ A )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  C ) )
4937, 48eqtrd 2475 . . 3  |-  ( ph  ->  ( ( j  e.  ( M ... N
)  |->  A )  o.  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  C ) )
5049oveq2d 6107 . 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 2475 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 1369    e. wcel 1756   F/_wnfc 2566   _Vcvv 2972   [_csb 3288    e. cmpt 4350    o. ccom 4844   ` cfv 5418  (class class class)co 6091   CCcc 9280    + caddc 9285    - cmin 9595   ZZcz 10646   ...cfz 11437   Basecbs 14174   0gc0g 14378    gsumg cgsu 14379  CMndccmn 16277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-0g 14380  df-gsum 14381  df-mnd 15415  df-cntz 15835  df-cmn 16279
This theorem is referenced by:  srgbinomlem4  16641
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