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Theorem fsfnn0gsumfsffz 16882
Description: Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019.)
Hypotheses
Ref Expression
nn0gsumfz.b  |-  B  =  ( Base `  G
)
nn0gsumfz.0  |-  .0.  =  ( 0g `  G )
nn0gsumfz.g  |-  ( ph  ->  G  e. CMnd )
nn0gsumfz.f  |-  ( ph  ->  F  e.  ( B  ^m  NN0 ) )
fsfnn0gsumfsffz.s  |-  ( ph  ->  S  e.  NN0 )
fsfnn0gsumfsffz.h  |-  H  =  ( F  |`  (
0 ... S ) )
Assertion
Ref Expression
fsfnn0gsumfsffz  |-  ( ph  ->  ( A. x  e. 
NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  )  ->  ( G  gsumg  F )  =  ( G 
gsumg  H ) ) )
Distinct variable groups:    x, F    x, S    x,  .0.
Allowed substitution hints:    ph( x)    B( x)    G( x)    H( x)

Proof of Theorem fsfnn0gsumfsffz
StepHypRef Expression
1 fsfnn0gsumfsffz.h . . . 4  |-  H  =  ( F  |`  (
0 ... S ) )
21oveq2i 6306 . . 3  |-  ( G 
gsumg  H )  =  ( G  gsumg  ( F  |`  (
0 ... S ) ) )
3 nn0gsumfz.b . . . 4  |-  B  =  ( Base `  G
)
4 nn0gsumfz.0 . . . 4  |-  .0.  =  ( 0g `  G )
5 nn0gsumfz.g . . . . 5  |-  ( ph  ->  G  e. CMnd )
65adantr 465 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  G  e. CMnd )
7 nn0ex 10813 . . . . 5  |-  NN0  e.  _V
87a1i 11 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  NN0  e.  _V )
9 nn0gsumfz.f . . . . . 6  |-  ( ph  ->  F  e.  ( B  ^m  NN0 ) )
10 elmapi 7452 . . . . . 6  |-  ( F  e.  ( B  ^m  NN0 )  ->  F : NN0
--> B )
119, 10syl 16 . . . . 5  |-  ( ph  ->  F : NN0 --> B )
1211adantr 465 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  F : NN0 --> B )
13 fvex 5882 . . . . . . 7  |-  ( 0g
`  G )  e. 
_V
144, 13eqeltri 2551 . . . . . 6  |-  .0.  e.  _V
1514a1i 11 . . . . 5  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  .0.  e.  _V )
169adantr 465 . . . . 5  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  F  e.  ( B  ^m  NN0 ) )
17 fsfnn0gsumfsffz.s . . . . . 6  |-  ( ph  ->  S  e.  NN0 )
1817adantr 465 . . . . 5  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  S  e.  NN0 )
19 simpr 461 . . . . 5  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )
2015, 16, 18, 19suppssfz 12080 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  ( F supp  .0.  )  C_  (
0 ... S ) )
21 elmapfun 7454 . . . . . . . . 9  |-  ( F  e.  ( B  ^m  NN0 )  ->  Fun  F )
229, 21syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  F )
2314a1i 11 . . . . . . . 8  |-  ( ph  ->  .0.  e.  _V )
249, 22, 233jca 1176 . . . . . . 7  |-  ( ph  ->  ( F  e.  ( B  ^m  NN0 )  /\  Fun  F  /\  .0.  e.  _V ) )
2524adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  C_  ( 0 ... S ) )  ->  ( F  e.  ( B  ^m  NN0 )  /\  Fun  F  /\  .0.  e.  _V ) )
26 fzfid 12063 . . . . . . 7  |-  ( ph  ->  ( 0 ... S
)  e.  Fin )
2726anim1i 568 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  C_  ( 0 ... S ) )  ->  ( ( 0 ... S )  e. 
Fin  /\  ( F supp  .0.  )  C_  ( 0 ... S ) ) )
28 suppssfifsupp 7856 . . . . . 6  |-  ( ( ( F  e.  ( B  ^m  NN0 )  /\  Fun  F  /\  .0.  e.  _V )  /\  (
( 0 ... S
)  e.  Fin  /\  ( F supp  .0.  )  C_  ( 0 ... S
) ) )  ->  F finSupp  .0.  )
2925, 27, 28syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  C_  ( 0 ... S ) )  ->  F finSupp  .0.  )
3020, 29syldan 470 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  F finSupp  .0.  )
313, 4, 6, 8, 12, 20, 30gsumres 16792 . . 3  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  ( G  gsumg  ( F  |`  (
0 ... S ) ) )  =  ( G 
gsumg  F ) )
322, 31syl5req 2521 . 2  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  ) )  ->  ( G  gsumg  F )  =  ( G  gsumg  H ) )
3332ex 434 1  |-  ( ph  ->  ( A. x  e. 
NN0  ( S  < 
x  ->  ( F `  x )  =  .0.  )  ->  ( G  gsumg  F )  =  ( G 
gsumg  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    C_ wss 3481   class class class wbr 4453    |` cres 5007   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   supp csupp 6913    ^m cmap 7432   Fincfn 7528   finSupp cfsupp 7841   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-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:  nn0gsumfz  16883  gsummptnn0fz  16885
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