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Theorem gsummptshft 17157
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 6298 . . . 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 2454 . . . 4  |-  ( j  e.  ( M ... N )  |->  A )  =  ( j  e.  ( M ... N
)  |->  A )
86, 7fmptd 6031 . . 3  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A ) : ( M ... N ) --> B )
9 fzfid 12068 . . . 4  |-  ( ph  ->  ( M ... N
)  e.  Fin )
10 fvex 5858 . . . . . 6  |-  ( 0g
`  G )  e. 
_V
112, 10eqeltri 2538 . . . . 5  |-  .0.  e.  _V
1211a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
137, 9, 6, 12fsuppmptdm 7832 . . 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 13678 . . 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 17126 . 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 11691 . . . . . . . 8  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  k  e.  ZZ )
2019zcnd 10966 . . . . . . 7  |-  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  ->  k  e.  CC )
2114zcnd 10966 . . . . . . 7  |-  ( ph  ->  K  e.  CC )
22 npcan 9820 . . . . . . 7  |-  ( ( k  e.  CC  /\  K  e.  CC )  ->  ( ( k  -  K )  +  K
)  =  k )
2320, 21, 22syl2anr 476 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( k  -  K
)  +  K )  =  k )
24 simpr 459 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )
2523, 24eqeltrd 2542 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( k  -  K
)  +  K )  e.  ( ( M  +  K ) ... ( N  +  K
) ) )
2615, 16jca 530 . . . . . . 7  |-  ( ph  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2726adantr 463 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
2819adantl 464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  k  e.  ZZ )
2914adantr 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  K  e.  ZZ )
3028, 29zsubcld 10970 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  ZZ )
31 fzaddel 11722 . . . . . 6  |-  ( ( ( 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 1224 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
( k  -  K
)  e.  ( M ... N )  <->  ( (
k  -  K )  +  K )  e.  ( ( M  +  K ) ... ( N  +  K )
) ) )
3325, 32mpbird 232 . . . 4  |-  ( (
ph  /\  k  e.  ( ( M  +  K ) ... ( N  +  K )
) )  ->  (
k  -  K )  e.  ( M ... N ) )
34 eqidd 2455 . . . 4  |-  ( ph  ->  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K
) )  |->  ( k  -  K ) ) )
35 eqidd 2455 . . . 4  |-  ( ph  ->  ( j  e.  ( M ... N ) 
|->  A )  =  ( j  e.  ( M ... N )  |->  A ) )
36 gsummptshft.c . . . 4  |-  ( j  =  ( k  -  K )  ->  A  =  C )
3733, 34, 35, 36fmptco 6040 . . 3  |-  ( ph  ->  ( ( j  e.  ( M ... N
)  |->  A )  o.  ( k  e.  ( ( M  +  K
) ... ( N  +  K ) )  |->  ( k  -  K ) ) )  =  ( k  e.  ( ( M  +  K ) ... ( N  +  K ) )  |->  C ) )
3837oveq2d 6286 . 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 ) ) )
3918, 38eqtrd 2495 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    |-> cmpt 4497    o. ccom 4992   ` cfv 5570  (class class class)co 6270   CCcc 9479    + caddc 9484    - cmin 9796   ZZcz 10860   ...cfz 11675   Basecbs 14719   0gc0g 14932    gsumg cgsu 14933  CMndccmn 17000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-cntz 16557  df-cmn 17002
This theorem is referenced by:  srgbinomlem4  17392  cpmadugsumlemF  19547
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