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Theorem nn0gsumfz 17139
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 ) )
nn0gsumfz.y  |-  ( ph  ->  F finSupp  .0.  )
Assertion
Ref Expression
nn0gsumfz  |-  ( ph  ->  E. s  e.  NN0  E. f  e.  ( B  ^m  ( 0 ... s ) ) ( f  =  ( F  |`  ( 0 ... s
) )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  /\  ( G  gsumg  F )  =  ( G  gsumg  f ) ) )
Distinct variable groups:    B, f    f, F, s, x    f, G    .0. , f, s, x    ph, f, s
Allowed substitution hints:    ph( x)    B( x, s)    G( x, s)

Proof of Theorem nn0gsumfz
StepHypRef Expression
1 nn0gsumfz.f . . . 4  |-  ( ph  ->  F  e.  ( B  ^m  NN0 ) )
2 nn0gsumfz.0 . . . . 5  |-  .0.  =  ( 0g `  G )
3 fvex 5882 . . . . 5  |-  ( 0g
`  G )  e. 
_V
42, 3eqeltri 2541 . . . 4  |-  .0.  e.  _V
51, 4jctir 538 . . 3  |-  ( ph  ->  ( F  e.  ( B  ^m  NN0 )  /\  .0.  e.  _V )
)
6 nn0gsumfz.y . . 3  |-  ( ph  ->  F finSupp  .0.  )
7 fsuppmapnn0ub 12104 . . 3  |-  ( ( F  e.  ( B  ^m  NN0 )  /\  .0.  e.  _V )  -> 
( F finSupp  .0.  ->  E. s  e.  NN0  A. x  e.  NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  ) ) )
85, 6, 7sylc 60 . 2  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  ( F `  x )  =  .0.  ) )
9 eqidd 2458 . . . . 5  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  -> 
( F  |`  (
0 ... s ) )  =  ( F  |`  ( 0 ... s
) ) )
10 simpr 461 . . . . 5  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  ->  A. x  e.  NN0  ( s  <  x  ->  ( F `  x
)  =  .0.  )
)
11 nn0gsumfz.b . . . . . . 7  |-  B  =  ( Base `  G
)
12 nn0gsumfz.g . . . . . . . 8  |-  ( ph  ->  G  e. CMnd )
1312adantr 465 . . . . . . 7  |-  ( (
ph  /\  s  e.  NN0 )  ->  G  e. CMnd )
141adantr 465 . . . . . . 7  |-  ( (
ph  /\  s  e.  NN0 )  ->  F  e.  ( B  ^m  NN0 )
)
15 simpr 461 . . . . . . 7  |-  ( (
ph  /\  s  e.  NN0 )  ->  s  e.  NN0 )
16 eqid 2457 . . . . . . 7  |-  ( F  |`  ( 0 ... s
) )  =  ( F  |`  ( 0 ... s ) )
1711, 2, 13, 14, 15, 16fsfnn0gsumfsffz 17138 . . . . . 6  |-  ( (
ph  /\  s  e.  NN0 )  ->  ( A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) ) )
1817imp 429 . . . . 5  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) )
1914adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  ->  F  e.  ( B  ^m  NN0 ) )
20 fzssuz 11750 . . . . . . . 8  |-  ( 0 ... s )  C_  ( ZZ>= `  0 )
21 nn0uz 11140 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
2220, 21sseqtr4i 3532 . . . . . . 7  |-  ( 0 ... s )  C_  NN0
23 elmapssres 7462 . . . . . . 7  |-  ( ( F  e.  ( B  ^m  NN0 )  /\  ( 0 ... s
)  C_  NN0 )  -> 
( F  |`  (
0 ... s ) )  e.  ( B  ^m  ( 0 ... s
) ) )
2419, 22, 23sylancl 662 . . . . . 6  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  -> 
( F  |`  (
0 ... s ) )  e.  ( B  ^m  ( 0 ... s
) ) )
25 eqeq1 2461 . . . . . . . 8  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  (
f  =  ( F  |`  ( 0 ... s
) )  <->  ( F  |`  ( 0 ... s
) )  =  ( F  |`  ( 0 ... s ) ) ) )
26 oveq2 6304 . . . . . . . . 9  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) )
2726eqeq2d 2471 . . . . . . . 8  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  ( F  |`  ( 0 ... s ) ) ) ) )
2825, 273anbi13d 1301 . . . . . . 7  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  (
( f  =  ( F  |`  ( 0 ... s ) )  /\  A. x  e. 
NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  )  /\  ( G 
gsumg  F )  =  ( G  gsumg  f ) )  <->  ( ( F  |`  ( 0 ... s ) )  =  ( F  |`  (
0 ... s ) )  /\  A. x  e. 
NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  )  /\  ( G 
gsumg  F )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) ) ) )
2928adantl 466 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  (
s  <  x  ->  ( F `  x )  =  .0.  ) )  /\  f  =  ( F  |`  ( 0 ... s ) ) )  ->  ( (
f  =  ( F  |`  ( 0 ... s
) )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  /\  ( G  gsumg  F )  =  ( G  gsumg  f ) )  <->  ( ( F  |`  ( 0 ... s ) )  =  ( F  |`  (
0 ... s ) )  /\  A. x  e. 
NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  )  /\  ( G 
gsumg  F )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) ) ) )
3024, 29rspcedv 3214 . . . . 5  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  -> 
( ( ( F  |`  ( 0 ... s
) )  =  ( F  |`  ( 0 ... s ) )  /\  A. x  e. 
NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  )  /\  ( G 
gsumg  F )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) )  ->  E. f  e.  ( B  ^m  (
0 ... s ) ) ( f  =  ( F  |`  ( 0 ... s ) )  /\  A. x  e. 
NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  )  /\  ( G 
gsumg  F )  =  ( G  gsumg  f ) ) ) )
319, 10, 18, 30mp3and 1327 . . . 4  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  ) )  ->  E. f  e.  ( B  ^m  ( 0 ... s ) ) ( f  =  ( F  |`  ( 0 ... s
) )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  /\  ( G  gsumg  F )  =  ( G  gsumg  f ) ) )
3231ex 434 . . 3  |-  ( (
ph  /\  s  e.  NN0 )  ->  ( A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  ->  E. f  e.  ( B  ^m  (
0 ... s ) ) ( f  =  ( F  |`  ( 0 ... s ) )  /\  A. x  e. 
NN0  ( s  < 
x  ->  ( F `  x )  =  .0.  )  /\  ( G 
gsumg  F )  =  ( G  gsumg  f ) ) ) )
3332reximdva 2932 . 2  |-  ( ph  ->  ( E. s  e. 
NN0  A. x  e.  NN0  ( s  <  x  ->  ( F `  x
)  =  .0.  )  ->  E. s  e.  NN0  E. f  e.  ( B  ^m  ( 0 ... s ) ) ( f  =  ( F  |`  ( 0 ... s
) )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  /\  ( G  gsumg  F )  =  ( G  gsumg  f ) ) ) )
348, 33mpd 15 1  |-  ( ph  ->  E. s  e.  NN0  E. f  e.  ( B  ^m  ( 0 ... s ) ) ( f  =  ( F  |`  ( 0 ... s
) )  /\  A. x  e.  NN0  ( s  <  x  ->  ( F `  x )  =  .0.  )  /\  ( G  gsumg  F )  =  ( G  gsumg  f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    C_ wss 3471   class class class wbr 4456    |` cres 5010   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   finSupp cfsupp 7847   0cc0 9509    < clt 9645   NN0cn0 10816   ZZ>=cuz 11106   ...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-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:  nn0gsumfz0  17140
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