MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsummptnn0fz Structured version   Unicode version

Theorem gsummptnn0fz 17141
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 1708 . . . . 5  |-  F/ x
( S  <  k  ->  C  =  .0.  )
3 nfv 1708 . . . . . 6  |-  F/ k  S  <  x
4 nfcsb1v 3446 . . . . . . 7  |-  F/_ k [_ x  /  k ]_ C
54nfeq1 2634 . . . . . 6  |-  F/ k
[_ x  /  k ]_ C  =  .0.
63, 5nfim 1921 . . . . 5  |-  F/ k ( S  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
7 breq2 4460 . . . . . 6  |-  ( k  =  x  ->  ( S  <  k  <->  S  <  x ) )
8 csbeq1a 3439 . . . . . . 7  |-  ( k  =  x  ->  C  =  [_ x  /  k ]_ C )
98eqeq1d 2459 . . . . . 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 3080 . . . 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 2457 . . . . . . . . 9  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
2221fvmpts 5958 . . . . . . . 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 2498 . . . . . 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 2865 . . 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 2541 . . . . . 6  |-  B  e. 
_V
37 nn0ex 10822 . . . . . 6  |-  NN0  e.  _V
3836, 37pm3.2i 455 . . . . 5  |-  ( B  e.  _V  /\  NN0  e.  _V )
39 elmapg 7451 . . . . 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 11797 . . . . . 6  |-  ( k  e.  ( 0 ... S )  ->  k  e.  NN0 )
4443ssriv 3503 . . . . 5  |-  ( 0 ... S )  C_  NN0
45 resmpt 5333 . . . . 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 17138 . 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 1395   F/wnf 1617    e. wcel 1819   A.wral 2807   _Vcvv 3109   [_csb 3430    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515    |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   0cc0 9509    < clt 9645   NN0cn0 10816   ...cfz 11697   Basecbs 14644   0gc0g 14857    gsumg cgsu 14858  CMndccmn 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-0g 14859  df-gsum 14860  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-cntz 16482  df-cmn 16927
This theorem is referenced by:  gsummptnn0fzv  17142  gsummoncoe1  18473  pmatcollpwfi  19410
  Copyright terms: Public domain W3C validator