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Theorem nn0gsumfz 30804
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 5701 . . . . 5  |-  ( 0g
`  G )  e. 
_V
42, 3eqeltri 2513 . . . 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 30796 . . 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 2444 . . . . 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 2443 . . . . . . 7  |-  ( F  |`  ( 0 ... s
) )  =  ( F  |`  ( 0 ... s ) )
1711, 2, 13, 14, 15, 16fsfnn0gsumfsffz 30803 . . . . . 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 11499 . . . . . . . 8  |-  ( 0 ... s )  C_  ( ZZ>= `  0 )
21 nn0uz 10895 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
2220, 21sseqtr4i 3389 . . . . . . 7  |-  ( 0 ... s )  C_  NN0
23 elmapssres 7237 . . . . . . 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 2449 . . . . . . . 8  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  (
f  =  ( F  |`  ( 0 ... s
) )  <->  ( F  |`  ( 0 ... s
) )  =  ( F  |`  ( 0 ... s ) ) ) )
26 oveq2 6099 . . . . . . . . 9  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( F  |`  (
0 ... s ) ) ) )
2726eqeq2d 2454 . . . . . . . 8  |-  ( f  =  ( F  |`  ( 0 ... s
) )  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  ( F  |`  ( 0 ... s ) ) ) ) )
2825, 273anbi13d 1291 . . . . . . 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 3077 . . . . 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 1317 . . . 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 2828 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972    C_ wss 3328   class class class wbr 4292    |` cres 4842   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   finSupp cfsupp 7620   0cc0 9282    < clt 9418   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437   Basecbs 14174   0gc0g 14378    gsumg cgsu 14379  CMndccmn 16277
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-0g 14380  df-gsum 14381  df-mnd 15415  df-cntz 15835  df-cmn 16279
This theorem is referenced by:  nn0gsumfz0  30805
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