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Theorem gsummptnn0fz 30811
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 1673 . . . . 5  |-  F/ x
( S  <  k  ->  C  =  .0.  )
3 nfv 1673 . . . . . 6  |-  F/ k  S  <  x
4 nfcsb1v 3309 . . . . . . 7  |-  F/_ k [_ x  /  k ]_ C
54nfeq1 2593 . . . . . 6  |-  F/ k
[_ x  /  k ]_ C  =  .0.
63, 5nfim 1853 . . . . 5  |-  F/ k ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
7 breq2 4301 . . . . . 6  |-  ( k  =  x  ->  ( S  <  k  <->  S  <  x ) )
8 csbeq1a 3302 . . . . . . 7  |-  ( k  =  x  ->  C  =  [_ x  /  k ]_ C )
98eqeq1d 2451 . . . . . 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 2948 . . . 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 3710 . . . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
2221fvmpts 5781 . . . . . . . 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 2475 . . . . . 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 2799 . . 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 5869 . . . . 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 5706 . . . . . . 7  |-  ( Base `  G )  e.  _V
3630, 35eqeltri 2513 . . . . . 6  |-  B  e. 
_V
37 nn0ex 10590 . . . . . 6  |-  NN0  e.  _V
3836, 37pm3.2i 455 . . . . 5  |-  ( B  e.  _V  /\  NN0  e.  _V )
39 elmapg 7232 . . . . 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 11486 . . . . . 6  |-  ( k  e.  ( 0 ... S )  ->  k  e.  NN0 )
4443ssriv 3365 . . . . 5  |-  ( 0 ... S )  C_  NN0
45 resmpt 5161 . . . . 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 2447 . . 3  |-  ( k  e.  ( 0 ... S )  |->  C )  =  ( ( k  e.  NN0  |->  C )  |`  ( 0 ... S
) )
4830, 31, 32, 41, 42, 47fsfnn0gsumfsffz 30808 . 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 1369   F/wnf 1589    e. wcel 1756   A.wral 2720   _Vcvv 2977   [_csb 3293    C_ wss 3333   class class class wbr 4297    e. cmpt 4355    |` cres 4847   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   0cc0 9287    < clt 9423   NN0cn0 10584   ...cfz 11442   Basecbs 14179   0gc0g 14383    gsumg cgsu 14384  CMndccmn 16282
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-0g 14385  df-gsum 14386  df-mnd 15420  df-cntz 15840  df-cmn 16284
This theorem is referenced by:  gsummptnn0fzv  30812  gsummoncoe1  30848
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