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Theorem fsuppmapnn0fiub0 12155
Description: If all functions of a finite set of functions over the nonnegative integers are finitely supported, then all these functions are zero for all integers greater than a fixed integer. (Contributed by AV, 3-Oct-2019.)
Assertion
Ref Expression
fsuppmapnn0fiub0  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  ( A. f  e.  M  f finSupp  Z  ->  E. m  e.  NN0  A. f  e.  M  A. x  e. 
NN0  ( m  < 
x  ->  ( f `  x )  =  Z ) ) )
Distinct variable groups:    f, M, m    R, f, m    f, V, m    f, Z, m   
x, M    x, R    x, V    x, Z, f, m

Proof of Theorem fsuppmapnn0fiub0
StepHypRef Expression
1 fsuppmapnn0fiubex 12154 . 2  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  ( A. f  e.  M  f finSupp  Z  ->  E. m  e.  NN0  A. f  e.  M  ( f supp  Z
)  C_  ( 0 ... m ) ) )
2 ssel2 3402 . . . . . . . . . . . . . 14  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  f  e.  ( R  ^m  NN0 ) )
32ancoms 454 . . . . . . . . . . . . 13  |-  ( ( f  e.  M  /\  M  C_  ( R  ^m  NN0 ) )  ->  f  e.  ( R  ^m  NN0 ) )
4 elmapfn 7449 . . . . . . . . . . . . 13  |-  ( f  e.  ( R  ^m  NN0 )  ->  f  Fn  NN0 )
53, 4syl 17 . . . . . . . . . . . 12  |-  ( ( f  e.  M  /\  M  C_  ( R  ^m  NN0 ) )  ->  f  Fn  NN0 )
65expcom 436 . . . . . . . . . . 11  |-  ( M 
C_  ( R  ^m  NN0 )  ->  ( f  e.  M  ->  f  Fn 
NN0 ) )
763ad2ant1 1026 . . . . . . . . . 10  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
f  e.  M  -> 
f  Fn  NN0 )
)
87adantr 466 . . . . . . . . 9  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  ->  (
f  e.  M  -> 
f  Fn  NN0 )
)
98imp 430 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  f  Fn  NN0 )
10 nn0ex 10826 . . . . . . . . 9  |-  NN0  e.  _V
1110a1i 11 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  NN0  e.  _V )
12 simpll3 1046 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  Z  e.  V )
13 suppvalfn 6876 . . . . . . . 8  |-  ( ( f  Fn  NN0  /\  NN0 
e.  _V  /\  Z  e.  V )  ->  (
f supp  Z )  =  {
x  e.  NN0  | 
( f `  x
)  =/=  Z }
)
149, 11, 12, 13syl3anc 1264 . . . . . . 7  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  ( f supp  Z )  =  { x  e.  NN0  |  ( f `
 x )  =/= 
Z } )
1514sseq1d 3434 . . . . . 6  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  ( (
f supp  Z )  C_  (
0 ... m )  <->  { x  e.  NN0  |  ( f `
 x )  =/= 
Z }  C_  (
0 ... m ) ) )
16 rabss 3481 . . . . . 6  |-  ( { x  e.  NN0  | 
( f `  x
)  =/=  Z }  C_  ( 0 ... m
)  <->  A. x  e.  NN0  ( ( f `  x )  =/=  Z  ->  x  e.  ( 0 ... m ) ) )
1715, 16syl6bb 264 . . . . 5  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  ( (
f supp  Z )  C_  (
0 ... m )  <->  A. x  e.  NN0  ( ( f `
 x )  =/= 
Z  ->  x  e.  ( 0 ... m
) ) ) )
18 nne 2605 . . . . . . . . . 10  |-  ( -.  ( f `  x
)  =/=  Z  <->  ( f `  x )  =  Z )
1918biimpi 197 . . . . . . . . 9  |-  ( -.  ( f `  x
)  =/=  Z  -> 
( f `  x
)  =  Z )
20192a1d 27 . . . . . . . 8  |-  ( -.  ( f `  x
)  =/=  Z  -> 
( ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M )  /\  x  e.  NN0 )  ->  (
m  <  x  ->  ( f `  x )  =  Z ) ) )
21 elfz2nn0 11836 . . . . . . . . 9  |-  ( x  e.  ( 0 ... m )  <->  ( x  e.  NN0  /\  m  e. 
NN0  /\  x  <_  m ) )
22 nn0re 10829 . . . . . . . . . . . . 13  |-  ( x  e.  NN0  ->  x  e.  RR )
23 nn0re 10829 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  ->  m  e.  RR )
24 lenlt 9663 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  m  e.  RR )  ->  ( x  <_  m  <->  -.  m  <  x ) )
2522, 23, 24syl2an 479 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  m  e.  NN0 )  -> 
( x  <_  m  <->  -.  m  <  x ) )
26 pm2.21 111 . . . . . . . . . . . 12  |-  ( -.  m  <  x  -> 
( m  <  x  ->  ( f `  x
)  =  Z ) )
2725, 26syl6bi 231 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  m  e.  NN0 )  -> 
( x  <_  m  ->  ( m  <  x  ->  ( f `  x
)  =  Z ) ) )
28273impia 1202 . . . . . . . . . 10  |-  ( ( x  e.  NN0  /\  m  e.  NN0  /\  x  <_  m )  ->  (
m  <  x  ->  ( f `  x )  =  Z ) )
2928a1d 26 . . . . . . . . 9  |-  ( ( x  e.  NN0  /\  m  e.  NN0  /\  x  <_  m )  ->  (
( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M )  /\  x  e.  NN0 )  ->  (
m  <  x  ->  ( f `  x )  =  Z ) ) )
3021, 29sylbi 198 . . . . . . . 8  |-  ( x  e.  ( 0 ... m )  ->  (
( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M )  /\  x  e.  NN0 )  ->  (
m  <  x  ->  ( f `  x )  =  Z ) ) )
3120, 30ja 164 . . . . . . 7  |-  ( ( ( f `  x
)  =/=  Z  ->  x  e.  ( 0 ... m ) )  ->  ( ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  /\  x  e.  NN0 )  ->  ( m  <  x  ->  ( f `  x )  =  Z ) ) )
3231com12 32 . . . . . 6  |-  ( ( ( ( ( M 
C_  ( R  ^m  NN0 )  /\  M  e. 
Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M )  /\  x  e.  NN0 )  ->  (
( ( f `  x )  =/=  Z  ->  x  e.  ( 0 ... m ) )  ->  ( m  < 
x  ->  ( f `  x )  =  Z ) ) )
3332ralimdva 2773 . . . . 5  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  ( A. x  e.  NN0  ( ( f `  x )  =/=  Z  ->  x  e.  ( 0 ... m
) )  ->  A. x  e.  NN0  ( m  < 
x  ->  ( f `  x )  =  Z ) ) )
3417, 33sylbid 218 . . . 4  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  ( (
f supp  Z )  C_  (
0 ... m )  ->  A. x  e.  NN0  ( m  <  x  -> 
( f `  x
)  =  Z ) ) )
3534ralimdva 2773 . . 3  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  ->  ( A. f  e.  M  ( f supp  Z )  C_  ( 0 ... m
)  ->  A. f  e.  M  A. x  e.  NN0  ( m  < 
x  ->  ( f `  x )  =  Z ) ) )
3635reximdva 2839 . 2  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  ( E. m  e.  NN0  A. f  e.  M  ( f supp  Z )  C_  ( 0 ... m
)  ->  E. m  e.  NN0  A. f  e.  M  A. x  e. 
NN0  ( m  < 
x  ->  ( f `  x )  =  Z ) ) )
371, 36syld 45 1  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  ( A. f  e.  M  f finSupp  Z  ->  E. m  e.  NN0  A. f  e.  M  A. x  e. 
NN0  ( m  < 
x  ->  ( f `  x )  =  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 3022    C_ wss 3379   class class class wbr 4366    Fn wfn 5539   ` cfv 5544  (class class class)co 6249   supp csupp 6869    ^m cmap 7427   Fincfn 7524   finSupp cfsupp 7836   RRcr 9489   0cc0 9490    < clt 9626    <_ cle 9627   NN0cn0 10820   ...cfz 11735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-sup 7909  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736
This theorem is referenced by:  pmatcoe1fsupp  19667
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