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Theorem gsummptnn0fz 17558
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 1755 . . . . 5  |-  F/ x
( S  <  k  ->  C  =  .0.  )
3 nfv 1755 . . . . . 6  |-  F/ k  S  <  x
4 nfcsb1v 3354 . . . . . . 7  |-  F/_ k [_ x  /  k ]_ C
54nfeq1 2582 . . . . . 6  |-  F/ k
[_ x  /  k ]_ C  =  .0.
63, 5nfim 1980 . . . . 5  |-  F/ k ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
7 breq2 4370 . . . . . 6  |-  ( k  =  x  ->  ( S  <  k  <->  S  <  x ) )
8 csbeq1a 3347 . . . . . . 7  |-  ( k  =  x  ->  C  =  [_ x  /  k ]_ C )
98eqeq1d 2430 . . . . . 6  |-  ( k  =  x  ->  ( C  =  .0.  <->  [_ x  / 
k ]_ C  =  .0.  ) )
107, 9imbi12d 321 . . . . 5  |-  ( k  =  x  ->  (
( S  <  k  ->  C  =  .0.  )  <->  ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) ) )
112, 6, 10cbvral 2992 . . . 4  |-  ( A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )  <->  A. x  e.  NN0  ( S  < 
x  ->  [_ x  / 
k ]_ C  =  .0.  ) )
121, 11sylib 199 . . 3  |-  ( ph  ->  A. x  e.  NN0  ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) )
13 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  NN0 )  ->  x  e.  NN0 )
14 gsummptnn0fz.f . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
1514anim2i 571 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  ph )  ->  ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B
) )
1615ancoms 454 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B
) )
17 rspcsbela 3768 . . . . . . . . . . 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 534 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  NN0 )  ->  ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B ) )
2019adantr 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )
)
21 eqid 2428 . . . . . . . . 9  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
2221fvmpts 5911 . . . . . . . 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 462 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  [_ x  /  k ]_ C  =  .0.  )
2523, 24eqtrd 2462 . . . . . 6  |-  ( ( ( ph  /\  x  e.  NN0 )  /\  [_ x  /  k ]_ C  =  .0.  )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
2625ex 435 . . . . 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 2773 . . 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 6002 . . . . 5  |-  ( A. k  e.  NN0  C  e.  B  <->  ( k  e. 
NN0  |->  C ) : NN0 --> B )
3414, 33sylib 199 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  C ) : NN0 --> B )
35 fvex 5835 . . . . . . 7  |-  ( Base `  G )  e.  _V
3630, 35eqeltri 2502 . . . . . 6  |-  B  e. 
_V
37 nn0ex 10826 . . . . . 6  |-  NN0  e.  _V
3836, 37pm3.2i 456 . . . . 5  |-  ( B  e.  _V  /\  NN0  e.  _V )
39 elmapg 7440 . . . . 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 235 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  ( B  ^m  NN0 )
)
42 gsummptnn0fz.s . . 3  |-  ( ph  ->  S  e.  NN0 )
43 elfznn0 11838 . . . . . 6  |-  ( k  e.  ( 0 ... S )  ->  k  e.  NN0 )
4443ssriv 3411 . . . . 5  |-  ( 0 ... S )  C_  NN0
45 resmpt 5116 . . . . 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 2437 . . 3  |-  ( k  e.  ( 0 ... S )  |->  C )  =  ( ( k  e.  NN0  |->  C )  |`  ( 0 ... S
) )
4830, 31, 32, 41, 42, 47fsfnn0gsumfsffz 17555 . 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 187    /\ wa 370    = wceq 1437   F/wnf 1661    e. wcel 1872   A.wral 2714   _Vcvv 3022   [_csb 3338    C_ wss 3379   class class class wbr 4366    |-> cmpt 4425    |` cres 4798   -->wf 5540   ` cfv 5544  (class class class)co 6249    ^m cmap 7427   0cc0 9490    < clt 9626   NN0cn0 10820   ...cfz 11735   Basecbs 15064   0gc0g 15281    gsumg cgsu 15282  CMndccmn 17373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-oi 7978  df-card 8325  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-fzo 11867  df-seq 12164  df-hash 12466  df-0g 15283  df-gsum 15284  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-cntz 16914  df-cmn 17375
This theorem is referenced by:  gsummptnn0fzv  17559  gsummoncoe1  18841  pmatcollpwfi  19748
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