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Theorem gsummptnn0fz 16885
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 1683 . . . . 5  |-  F/ x
( S  <  k  ->  C  =  .0.  )
3 nfv 1683 . . . . . 6  |-  F/ k  S  <  x
4 nfcsb1v 3456 . . . . . . 7  |-  F/_ k [_ x  /  k ]_ C
54nfeq1 2644 . . . . . 6  |-  F/ k
[_ x  /  k ]_ C  =  .0.
63, 5nfim 1867 . . . . 5  |-  F/ k ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
7 breq2 4457 . . . . . 6  |-  ( k  =  x  ->  ( S  <  k  <->  S  <  x ) )
8 csbeq1a 3449 . . . . . . 7  |-  ( k  =  x  ->  C  =  [_ x  /  k ]_ C )
98eqeq1d 2469 . . . . . 6  |-  ( k  =  x  ->  ( C  =  .0.  <->  [_ x  / 
k ]_ C  =  .0.  ) )
107, 9imbi12d 320 . . . . 5  |-  ( k  =  x  ->  (
( S  <  k  ->  C  =  .0.  )  <->  ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) ) )
112, 6, 10cbvral 3089 . . . 4  |-  ( A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )  <->  A. x  e.  NN0  ( S  < 
x  ->  [_ x  / 
k ]_ C  =  .0.  ) )
121, 11sylib 196 . . 3  |-  ( ph  ->  A. x  e.  NN0  ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) )
13 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  x  e.  NN0 )
14 gsummptnn0fz.f . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
1514anim2i 569 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  ph )  ->  ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B
) )
1615ancoms 453 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B
) )
17 rspcsbela 3858 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ x  /  k ]_ C  e.  B )
1816, 17syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  [_ x  / 
k ]_ C  e.  B
)
1913, 18jca 532 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B ) )
2019adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )
)
21 eqid 2467 . . . . . . . . 9  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
2221fvmpts 5959 . . . . . . . 8  |-  ( ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )  ->  ( ( k  e. 
NN0  |->  C ) `  x )  =  [_ x  /  k ]_ C
)
2320, 22syl 16 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  [_ x  /  k ]_ C
)
24 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  [_ x  /  k ]_ C  =  .0.  )
2523, 24eqtrd 2508 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
2625ex 434 . . . . 5  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( [_ x  /  k ]_ C  =  .0.  ->  ( (
k  e.  NN0  |->  C ) `
 x )  =  .0.  ) )
2726imim2d 52 . . . 4  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )  ->  ( S  <  x  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
) )
2827ralimdva 2875 . . 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 6053 . . . . 5  |-  ( A. k  e.  NN0  C  e.  B  <->  ( k  e. 
NN0  |->  C ) : NN0 --> B )
3414, 33sylib 196 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  C ) : NN0 --> B )
35 fvex 5882 . . . . . . 7  |-  ( Base `  G )  e.  _V
3630, 35eqeltri 2551 . . . . . 6  |-  B  e. 
_V
37 nn0ex 10813 . . . . . 6  |-  NN0  e.  _V
3836, 37pm3.2i 455 . . . . 5  |-  ( B  e.  _V  /\  NN0  e.  _V )
39 elmapg 7445 . . . . 5  |-  ( ( B  e.  _V  /\  NN0 
e.  _V )  ->  (
( k  e.  NN0  |->  C )  e.  ( B  ^m  NN0 )  <->  ( k  e.  NN0  |->  C ) : NN0 --> B ) )
4038, 39mp1i 12 . . . 4  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C )  e.  ( B  ^m  NN0 ) 
<->  ( k  e.  NN0  |->  C ) : NN0 --> B ) )
4134, 40mpbird 232 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  ( B  ^m  NN0 )
)
42 gsummptnn0fz.s . . 3  |-  ( ph  ->  S  e.  NN0 )
43 elfznn0 11782 . . . . . 6  |-  ( k  e.  ( 0 ... S )  ->  k  e.  NN0 )
4443ssriv 3513 . . . . 5  |-  ( 0 ... S )  C_  NN0
45 resmpt 5329 . . . . 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 2480 . . 3  |-  ( k  e.  ( 0 ... S )  |->  C )  =  ( ( k  e.  NN0  |->  C )  |`  ( 0 ... S
) )
4830, 31, 32, 41, 42, 47fsfnn0gsumfsffz 16882 . 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 184    /\ wa 369    = wceq 1379   F/wnf 1599    e. wcel 1767   A.wral 2817   _Vcvv 3118   [_csb 3440    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   0cc0 9504    < clt 9640   NN0cn0 10807   ...cfz 11684   Basecbs 14506   0gc0g 14711    gsumg cgsu 14712  CMndccmn 16669
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-fal 1385  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-map 7434  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 14713  df-gsum 14714  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-cntz 16226  df-cmn 16671
This theorem is referenced by:  gsummptnn0fzv  16886  gsummoncoe1  18214  pmatcollpwfi  19150
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