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Theorem mptnn0fsupp 12085
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 2868 . . . . 5  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
3 eqid 2454 . . . . . 6  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
43fnmpt 5689 . . . . 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 3115 . . . . 5  |-  (  .0. 
e.  V  ->  .0.  e.  _V )
108, 9syl 16 . . . 4  |-  ( ph  ->  .0.  e.  _V )
11 suppvalfn 6898 . . . 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 1226 . . 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 2655 . . . . . . . . 9  |-  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
15 simpr 459 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
162ad2antrr 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  A. k  e.  NN0  C  e.  B
)
17 rspcsbela 3845 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ x  /  k ]_ C  e.  B )
1815, 16, 17syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  [_ x  /  k ]_ C  e.  B )
193fvmpts 5933 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )  ->  ( ( k  e. 
NN0  |->  C ) `  x )  =  [_ x  /  k ]_ C
)
2015, 18, 19syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  [_ x  /  k ]_ C
)
2120eqeq1d 2456 . . . . . . . . 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 314 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( s  <  x  ->  -.  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  )  <->  ( s  <  x  ->  [_ x  / 
k ]_ C  =  .0.  ) ) )
2423ralbidva 2890 . . . . . 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 2962 . . . . 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 12077 . . . 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 2542 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) supp  .0.  )  e.  Fin )
30 funmpt 5606 . . . 4  |-  Fun  (
k  e.  NN0  |->  C )
3130a1i 11 . . 3  |-  ( ph  ->  Fun  ( k  e. 
NN0  |->  C ) )
326mptex 6118 . . . 4  |-  ( k  e.  NN0  |->  C )  e.  _V
3332a1i 11 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  _V )
34 funisfsupp 7826 . . 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 1226 . 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   [_csb 3420   class class class wbr 4439    |-> cmpt 4497   Fun wfun 5564    Fn wfn 5565   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821    < clt 9617   NN0cn0 10791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676
This theorem is referenced by:  mptnn0fsuppd  12086  mptcoe1fsupp  18450  mptcoe1matfsupp  19470  pm2mp  19493  chfacffsupp  19524  chfacfscmulfsupp  19527  chfacfpmmulfsupp  19531  cayhamlem4  19556  ply1mulgsumlem3  33242  ply1mulgsumlem4  33243
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