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Theorem mptnn0fsupp 12066
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 2878 . . . . 5  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
3 eqid 2467 . . . . . 6  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
43fnmpt 5705 . . . . 5  |-  ( A. k  e.  NN0  C  e.  B  ->  ( k  e.  NN0  |->  C )  Fn 
NN0 )
52, 4syl 16 . . . 4  |-  ( ph  ->  ( k  e.  NN0  |->  C )  Fn  NN0 )
6 nn0ex 10797 . . . . 5  |-  NN0  e.  _V
76a1i 11 . . . 4  |-  ( ph  ->  NN0  e.  _V )
8 mptnn0fsupp.0 . . . . 5  |-  ( ph  ->  .0.  e.  V )
9 elex 3122 . . . . 5  |-  (  .0. 
e.  V  ->  .0.  e.  _V )
108, 9syl 16 . . . 4  |-  ( ph  ->  .0.  e.  _V )
11 suppvalfn 6905 . . . 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 1228 . . 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 2668 . . . . . . . . 9  |-  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
15 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
162ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  A. k  e.  NN0  C  e.  B
)
17 rspcsbela 3853 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ x  /  k ]_ C  e.  B )
1815, 16, 17syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  [_ x  /  k ]_ C  e.  B )
193fvmpts 5950 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )  ->  ( ( k  e. 
NN0  |->  C ) `  x )  =  [_ x  /  k ]_ C
)
2015, 18, 19syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  [_ x  /  k ]_ C
)
2120eqeq1d 2469 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( ( k  e. 
NN0  |->  C ) `  x )  =  .0.  <->  [_ x  /  k ]_ C  =  .0.  )
)
2214, 21syl5bb 257 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  [_ x  /  k ]_ C  =  .0.  ) )
2322imbi2d 316 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( s  <  x  ->  -.  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  )  <->  ( s  <  x  ->  [_ x  / 
k ]_ C  =  .0.  ) ) )
2423ralbidva 2900 . . . . . 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 2970 . . . . 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 232 . . . 4  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  ) )
27 rabssnn0fi 12058 . . . 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 212 . . 3  |-  ( ph  ->  { x  e.  NN0  |  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  }  e.  Fin )
2912, 28eqeltrd 2555 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) supp  .0.  )  e.  Fin )
30 funmpt 5622 . . . 4  |-  Fun  (
k  e.  NN0  |->  C )
3130a1i 11 . . 3  |-  ( ph  ->  Fun  ( k  e. 
NN0  |->  C ) )
326mptex 6129 . . . 4  |-  ( k  e.  NN0  |->  C )  e.  _V
3332a1i 11 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  _V )
34 funisfsupp 7830 . . 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 1228 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) finSupp  .0.  <->  (
( k  e.  NN0  |->  C ) supp  .0.  )  e.  Fin ) )
3629, 35mpbird 232 1  |-  ( ph  ->  ( k  e.  NN0  |->  C ) finSupp  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   [_csb 3435   class class class wbr 4447    |-> cmpt 4505   Fun wfun 5580    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   supp csupp 6898   Fincfn 7513   finSupp cfsupp 7825    < clt 9624   NN0cn0 10791
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669
This theorem is referenced by:  mptnn0fsuppd  12067  mptcoe1fsupp  18023  mptcoe1matfsupp  19067  pm2mp  19090  chfacffsupp  19121  chfacfscmulfsupp  19124  chfacfpmmulfsupp  19128  cayhamlem4  19153  ply1mulgsumlem3  32061  ply1mulgsumlem4  32062
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