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Theorem gsummptnn0fz 17663
Description: A final group sum over a function over the nonnegative integers (given as mapping) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.)
Hypotheses
Ref Expression
gsummptnn0fz.k  |-  F/ k
ph
gsummptnn0fz.b  |-  B  =  ( Base `  G
)
gsummptnn0fz.0  |-  .0.  =  ( 0g `  G )
gsummptnn0fz.g  |-  ( ph  ->  G  e. CMnd )
gsummptnn0fz.f  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
gsummptnn0fz.s  |-  ( ph  ->  S  e.  NN0 )
gsummptnn0fz.u  |-  ( ph  ->  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )
)
Assertion
Ref Expression
gsummptnn0fz  |-  ( ph  ->  ( G  gsumg  ( k  e.  NN0  |->  C ) )  =  ( G  gsumg  ( k  e.  ( 0 ... S ) 
|->  C ) ) )
Distinct variable groups:    B, k    S, k    .0. , k
Allowed substitution hints:    ph( k)    C( k)    G( k)

Proof of Theorem gsummptnn0fz
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 gsummptnn0fz.u . . . 4  |-  ( ph  ->  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )
)
2 nfv 1771 . . . . 5  |-  F/ x
( S  <  k  ->  C  =  .0.  )
3 nfv 1771 . . . . . 6  |-  F/ k  S  <  x
4 nfcsb1v 3390 . . . . . . 7  |-  F/_ k [_ x  /  k ]_ C
54nfeq1 2615 . . . . . 6  |-  F/ k
[_ x  /  k ]_ C  =  .0.
63, 5nfim 2013 . . . . 5  |-  F/ k ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
7 breq2 4419 . . . . . 6  |-  ( k  =  x  ->  ( S  <  k  <->  S  <  x ) )
8 csbeq1a 3383 . . . . . . 7  |-  ( k  =  x  ->  C  =  [_ x  /  k ]_ C )
98eqeq1d 2463 . . . . . 6  |-  ( k  =  x  ->  ( C  =  .0.  <->  [_ x  / 
k ]_ C  =  .0.  ) )
107, 9imbi12d 326 . . . . 5  |-  ( k  =  x  ->  (
( S  <  k  ->  C  =  .0.  )  <->  ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) ) )
112, 6, 10cbvral 3026 . . . 4  |-  ( A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )  <->  A. x  e.  NN0  ( S  < 
x  ->  [_ x  / 
k ]_ C  =  .0.  ) )
121, 11sylib 201 . . 3  |-  ( ph  ->  A. x  e.  NN0  ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) )
13 simpr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  x  e.  NN0 )
14 gsummptnn0fz.f . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
1514anim2i 577 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  ph )  ->  ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B
) )
1615ancoms 459 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B
) )
17 rspcsbela 3806 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ x  /  k ]_ C  e.  B )
1816, 17syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  [_ x  / 
k ]_ C  e.  B
)
1913, 18jca 539 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B ) )
2019adantr 471 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )
)
21 eqid 2461 . . . . . . . . 9  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
2221fvmpts 5973 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )  ->  ( ( k  e. 
NN0  |->  C ) `  x )  =  [_ x  /  k ]_ C
)
2320, 22syl 17 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  [_ x  /  k ]_ C
)
24 simpr 467 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  [_ x  /  k ]_ C  =  .0.  )
2523, 24eqtrd 2495 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
2625ex 440 . . . . 5  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( [_ x  /  k ]_ C  =  .0.  ->  ( (
k  e.  NN0  |->  C ) `
 x )  =  .0.  ) )
2726imim2d 54 . . . 4  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )  ->  ( S  <  x  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
) )
2827ralimdva 2807 . . 3  |-  ( ph  ->  ( A. x  e. 
NN0  ( S  < 
x  ->  [_ x  / 
k ]_ C  =  .0.  )  ->  A. x  e.  NN0  ( S  < 
x  ->  ( (
k  e.  NN0  |->  C ) `
 x )  =  .0.  ) ) )
2912, 28mpd 15 . 2  |-  ( ph  ->  A. x  e.  NN0  ( S  <  x  -> 
( ( k  e. 
NN0  |->  C ) `  x )  =  .0.  ) )
30 gsummptnn0fz.b . . 3  |-  B  =  ( Base `  G
)
31 gsummptnn0fz.0 . . 3  |-  .0.  =  ( 0g `  G )
32 gsummptnn0fz.g . . 3  |-  ( ph  ->  G  e. CMnd )
3321fmpt 6065 . . . . 5  |-  ( A. k  e.  NN0  C  e.  B  <->  ( k  e. 
NN0  |->  C ) : NN0 --> B )
3414, 33sylib 201 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  C ) : NN0 --> B )
35 fvex 5897 . . . . . . 7  |-  ( Base `  G )  e.  _V
3630, 35eqeltri 2535 . . . . . 6  |-  B  e. 
_V
37 nn0ex 10903 . . . . . 6  |-  NN0  e.  _V
3836, 37pm3.2i 461 . . . . 5  |-  ( B  e.  _V  /\  NN0  e.  _V )
39 elmapg 7510 . . . . 5  |-  ( ( B  e.  _V  /\  NN0 
e.  _V )  ->  (
( k  e.  NN0  |->  C )  e.  ( B  ^m  NN0 )  <->  ( k  e.  NN0  |->  C ) : NN0 --> B ) )
4038, 39mp1i 13 . . . 4  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C )  e.  ( B  ^m  NN0 ) 
<->  ( k  e.  NN0  |->  C ) : NN0 --> B ) )
4134, 40mpbird 240 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  ( B  ^m  NN0 )
)
42 gsummptnn0fz.s . . 3  |-  ( ph  ->  S  e.  NN0 )
43 elfznn0 11915 . . . . . 6  |-  ( k  e.  ( 0 ... S )  ->  k  e.  NN0 )
4443ssriv 3447 . . . . 5  |-  ( 0 ... S )  C_  NN0
45 resmpt 5172 . . . . 5  |-  ( ( 0 ... S ) 
C_  NN0  ->  ( ( k  e.  NN0  |->  C )  |`  ( 0 ... S
) )  =  ( k  e.  ( 0 ... S )  |->  C ) )
4644, 45ax-mp 5 . . . 4  |-  ( ( k  e.  NN0  |->  C )  |`  ( 0 ... S
) )  =  ( k  e.  ( 0 ... S )  |->  C )
4746eqcomi 2470 . . 3  |-  ( k  e.  ( 0 ... S )  |->  C )  =  ( ( k  e.  NN0  |->  C )  |`  ( 0 ... S
) )
4830, 31, 32, 41, 42, 47fsfnn0gsumfsffz 17660 . 2  |-  ( ph  ->  ( A. x  e. 
NN0  ( S  < 
x  ->  ( (
k  e.  NN0  |->  C ) `
 x )  =  .0.  )  ->  ( G  gsumg  ( k  e.  NN0  |->  C ) )  =  ( G  gsumg  ( k  e.  ( 0 ... S ) 
|->  C ) ) ) )
4929, 48mpd 15 1  |-  ( ph  ->  ( G  gsumg  ( k  e.  NN0  |->  C ) )  =  ( G  gsumg  ( k  e.  ( 0 ... S ) 
|->  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454   F/wnf 1677    e. wcel 1897   A.wral 2748   _Vcvv 3056   [_csb 3374    C_ wss 3415   class class class wbr 4415    |-> cmpt 4474    |` cres 4854   -->wf 5596   ` cfv 5600  (class class class)co 6314    ^m cmap 7497   0cc0 9564    < clt 9700   NN0cn0 10897   ...cfz 11812   Basecbs 15169   0gc0g 15386    gsumg cgsu 15387  CMndccmn 17478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-seq 12245  df-hash 12547  df-0g 15388  df-gsum 15389  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-cntz 17019  df-cmn 17480
This theorem is referenced by:  gsummptnn0fzv  17664  gsummoncoe1  18946  pmatcollpwfi  19854
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