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Theorem mptnn0fsupp 12240
Description: A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.)
Hypotheses
Ref Expression
mptnn0fsupp.0  |-  ( ph  ->  .0.  e.  V )
mptnn0fsupp.c  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  B )
mptnn0fsupp.s  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
)
Assertion
Ref Expression
mptnn0fsupp  |-  ( ph  ->  ( k  e.  NN0  |->  C ) finSupp  .0.  )
Distinct variable groups:    B, k    C, s, x    ph, k,
s, x    .0. , s, x
Allowed substitution hints:    B( x, s)    C( k)    V( x, k, s)    .0. ( k)

Proof of Theorem mptnn0fsupp
StepHypRef Expression
1 mptnn0fsupp.c . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  B )
21ralrimiva 2813 . . . . 5  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
3 eqid 2461 . . . . . 6  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
43fnmpt 5725 . . . . 5  |-  ( A. k  e.  NN0  C  e.  B  ->  ( k  e.  NN0  |->  C )  Fn 
NN0 )
52, 4syl 17 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  C )  Fn  NN0 )
6 nn0ex 10903 . . . . 5  |-  NN0  e.  _V
76a1i 11 . . . 4  |-  ( ph  ->  NN0  e.  _V )
8 mptnn0fsupp.0 . . . . 5  |-  ( ph  ->  .0.  e.  V )
9 elex 3065 . . . . 5  |-  (  .0. 
e.  V  ->  .0.  e.  _V )
108, 9syl 17 . . . 4  |-  ( ph  ->  .0.  e.  _V )
11 suppvalfn 6947 . . . 4  |-  ( ( ( k  e.  NN0  |->  C )  Fn  NN0  /\ 
NN0  e.  _V  /\  .0.  e.  _V )  ->  (
( k  e.  NN0  |->  C ) supp  .0.  )  =  { x  e.  NN0  |  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  } )
125, 7, 10, 11syl3anc 1276 . . 3  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) supp  .0.  )  =  { x  e.  NN0  |  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  } )
13 mptnn0fsupp.s . . . . 5  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
)
14 nne 2638 . . . . . . . . 9  |-  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
15 simpr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
162ad2antrr 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  A. k  e.  NN0  C  e.  B
)
17 rspcsbela 3806 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ x  /  k ]_ C  e.  B )
1815, 16, 17syl2anc 671 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  [_ x  /  k ]_ C  e.  B )
193fvmpts 5973 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )  ->  ( ( k  e. 
NN0  |->  C ) `  x )  =  [_ x  /  k ]_ C
)
2015, 18, 19syl2anc 671 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  [_ x  /  k ]_ C
)
2120eqeq1d 2463 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( ( k  e. 
NN0  |->  C ) `  x )  =  .0.  <->  [_ x  /  k ]_ C  =  .0.  )
)
2214, 21syl5bb 265 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  [_ x  /  k ]_ C  =  .0.  ) )
2322imbi2d 322 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( s  <  x  ->  -.  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  )  <->  ( s  <  x  ->  [_ x  / 
k ]_ C  =  .0.  ) ) )
2423ralbidva 2835 . . . . . 6  |-  ( (
ph  /\  s  e.  NN0 )  ->  ( A. x  e.  NN0  ( s  <  x  ->  -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  ) 
<-> 
A. x  e.  NN0  ( s  <  x  ->  [_ x  /  k ]_ C  =  .0.  ) ) )
2524rexbidva 2909 . . . . 5  |-  ( ph  ->  ( E. s  e. 
NN0  A. x  e.  NN0  ( s  <  x  ->  -.  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  )  <->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  [_ x  / 
k ]_ C  =  .0.  ) ) )
2613, 25mpbird 240 . . . 4  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  ) )
27 rabssnn0fi 12229 . . . 4  |-  ( { x  e.  NN0  | 
( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  }  e.  Fin  <->  E. s  e.  NN0  A. x  e. 
NN0  ( s  < 
x  ->  -.  (
( k  e.  NN0  |->  C ) `  x
)  =/=  .0.  )
)
2826, 27sylibr 217 . . 3  |-  ( ph  ->  { x  e.  NN0  |  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  }  e.  Fin )
2912, 28eqeltrd 2539 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) supp  .0.  )  e.  Fin )
30 funmpt 5636 . . . 4  |-  Fun  (
k  e.  NN0  |->  C )
3130a1i 11 . . 3  |-  ( ph  ->  Fun  ( k  e. 
NN0  |->  C ) )
326mptex 6160 . . . 4  |-  ( k  e.  NN0  |->  C )  e.  _V
3332a1i 11 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  _V )
34 funisfsupp 7913 . . 3  |-  ( ( Fun  ( k  e. 
NN0  |->  C )  /\  ( k  e.  NN0  |->  C )  e.  _V  /\  .0.  e.  _V )  ->  ( ( k  e. 
NN0  |->  C ) finSupp  .0.  <->  (
( k  e.  NN0  |->  C ) supp  .0.  )  e.  Fin ) )
3531, 33, 10, 34syl3anc 1276 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) finSupp  .0.  <->  (
( k  e.  NN0  |->  C ) supp  .0.  )  e.  Fin ) )
3629, 35mpbird 240 1  |-  ( ph  ->  ( k  e.  NN0  |->  C ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   E.wrex 2749   {crab 2752   _Vcvv 3056   [_csb 3374   class class class wbr 4415    |-> cmpt 4474   Fun wfun 5594    Fn wfn 5595   ` cfv 5600  (class class class)co 6314   supp csupp 6940   Fincfn 7594   finSupp cfsupp 7908    < clt 9700   NN0cn0 10897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813
This theorem is referenced by:  mptnn0fsuppd  12241  mptcoe1fsupp  18856  mptcoe1matfsupp  19874  pm2mp  19897  chfacffsupp  19928  chfacfscmulfsupp  19931  chfacfpmmulfsupp  19935  cayhamlem4  19960  ply1mulgsumlem3  40452  ply1mulgsumlem4  40453
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