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Theorem mptnn0fsupp 30814
Description: A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.)
Hypotheses
Ref Expression
mptnn0fsupp.0  |-  .0.  =  ( 0g `  R )
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    ph, k, s, x    C, s, x    .0. , s, x
Allowed substitution hints:    B( x, s)    C( k)    R( x, k, s)    .0. ( k)

Proof of Theorem mptnn0fsupp
StepHypRef Expression
1 mptnn0fsupp.c . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  B )
21ralrimiva 2811 . . . . 5  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
3 eqid 2443 . . . . . 6  |-  ( k  e.  NN0  |->  C )  =  ( k  e. 
NN0  |->  C )
43fnmpt 5549 . . . . 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 10597 . . . . 5  |-  NN0  e.  _V
76a1i 11 . . . 4  |-  ( ph  ->  NN0  e.  _V )
8 mptnn0fsupp.0 . . . . . 6  |-  .0.  =  ( 0g `  R )
9 fvex 5713 . . . . . 6  |-  ( 0g
`  R )  e. 
_V
108, 9eqeltri 2513 . . . . 5  |-  .0.  e.  _V
1110a1i 11 . . . 4  |-  ( ph  ->  .0.  e.  _V )
12 suppvalfn 6709 . . . 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.  } )
135, 7, 11, 12syl3anc 1218 . . 3  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) supp  .0.  )  =  { x  e.  NN0  |  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  } )
14 mptnn0fsupp.s . . . . 5  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  [_ x  /  k ]_ C  =  .0.  )
)
15 nne 2624 . . . . . . . . 9  |-  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  (
( k  e.  NN0  |->  C ) `  x
)  =  .0.  )
16 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  x  e.  NN0 )
172ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  A. k  e.  NN0  C  e.  B
)
18 rspcsbela 3717 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ x  /  k ]_ C  e.  B )
1916, 17, 18syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  [_ x  /  k ]_ C  e.  B )
203fvmpts 5788 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  [_ x  /  k ]_ C  e.  B )  ->  ( ( k  e. 
NN0  |->  C ) `  x )  =  [_ x  /  k ]_ C
)
2116, 19, 20syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( k  e.  NN0  |->  C ) `  x
)  =  [_ x  /  k ]_ C
)
2221eqeq1d 2451 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( ( k  e. 
NN0  |->  C ) `  x )  =  .0.  <->  [_ x  /  k ]_ C  =  .0.  )
)
2315, 22syl5bb 257 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  ( -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  <->  [_ x  /  k ]_ C  =  .0.  ) )
2423imbi2d 316 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  NN0 )  /\  x  e.  NN0 )  ->  (
( s  <  x  ->  -.  ( ( k  e.  NN0  |->  C ) `
 x )  =/= 
.0.  )  <->  ( s  <  x  ->  [_ x  / 
k ]_ C  =  .0.  ) ) )
2524ralbidva 2743 . . . . . 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.  ) ) )
2625rexbidva 2744 . . . . 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.  ) ) )
2714, 26mpbird 232 . . . 4  |-  ( ph  ->  E. s  e.  NN0  A. x  e.  NN0  (
s  <  x  ->  -.  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  ) )
28 rabssnn0fi 30759 . . . 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.  )
)
2927, 28sylibr 212 . . 3  |-  ( ph  ->  { x  e.  NN0  |  ( ( k  e. 
NN0  |->  C ) `  x )  =/=  .0.  }  e.  Fin )
3013, 29eqeltrd 2517 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) supp  .0.  )  e.  Fin )
31 funmpt 5466 . . . 4  |-  Fun  (
k  e.  NN0  |->  C )
3231a1i 11 . . 3  |-  ( ph  ->  Fun  ( k  e. 
NN0  |->  C ) )
336mptex 5960 . . . 4  |-  ( k  e.  NN0  |->  C )  e.  _V
3433a1i 11 . . 3  |-  ( ph  ->  ( k  e.  NN0  |->  C )  e.  _V )
35 funisfsupp 7637 . . 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 ) )
3632, 34, 11, 35syl3anc 1218 . 2  |-  ( ph  ->  ( ( k  e. 
NN0  |->  C ) finSupp  .0.  <->  (
( k  e.  NN0  |->  C ) supp  .0.  )  e.  Fin ) )
3730, 36mpbird 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 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   E.wrex 2728   {crab 2731   _Vcvv 2984   [_csb 3300   class class class wbr 4304    e. cmpt 4362   Fun wfun 5424    Fn wfn 5425   ` cfv 5430  (class class class)co 6103   supp csupp 6702   Fincfn 7322   finSupp cfsupp 7632    < clt 9430   NN0cn0 10591   0gc0g 14390
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  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-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450
This theorem is referenced by:  mptcoe1fsupp  30845  ply1mulgsumlem3  30858  ply1mulgsumlem4  30859  mptcoe1matfsupp  30903  pmattomply1lem  30920  pmattomply1mhmlem0  30939  pmattomply1mhmlem1  30940
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