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Theorem fsuppmapnn0fiub0 30798
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 30797 . 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 3349 . . . . . . . . . . . . . 14  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  f  e.  ( R  ^m  NN0 ) )
32ancoms 453 . . . . . . . . . . . . 13  |-  ( ( f  e.  M  /\  M  C_  ( R  ^m  NN0 ) )  ->  f  e.  ( R  ^m  NN0 ) )
4 elmapfn 7233 . . . . . . . . . . . . 13  |-  ( f  e.  ( R  ^m  NN0 )  ->  f  Fn  NN0 )
53, 4syl 16 . . . . . . . . . . . 12  |-  ( ( f  e.  M  /\  M  C_  ( R  ^m  NN0 ) )  ->  f  Fn  NN0 )
65expcom 435 . . . . . . . . . . 11  |-  ( M 
C_  ( R  ^m  NN0 )  ->  ( f  e.  M  ->  f  Fn 
NN0 ) )
763ad2ant1 1009 . . . . . . . . . 10  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
f  e.  M  -> 
f  Fn  NN0 )
)
87adantr 465 . . . . . . . . 9  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  ->  (
f  e.  M  -> 
f  Fn  NN0 )
)
98imp 429 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  f  Fn  NN0 )
10 nn0ex 10583 . . . . . . . . 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 1029 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  Z  e.  V )
13 suppvalfn 6695 . . . . . . . 8  |-  ( ( f  Fn  NN0  /\  NN0 
e.  _V  /\  Z  e.  V )  ->  (
f supp  Z )  =  {
x  e.  NN0  | 
( f `  x
)  =/=  Z }
)
149, 11, 12, 13syl3anc 1218 . . . . . . 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 3381 . . . . . 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 3427 . . . . . 6  |-  ( { x  e.  NN0  | 
( f `  x
)  =/=  Z }  C_  ( 0 ... m
)  <->  A. x  e.  NN0  ( ( f `  x )  =/=  Z  ->  x  e.  ( 0 ... m ) ) )
1715, 16syl6bb 261 . . . . 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 2610 . . . . . . . . . . 11  |-  ( -.  ( f `  x
)  =/=  Z  <->  ( f `  x )  =  Z )
1918biimpi 194 . . . . . . . . . 10  |-  ( -.  ( f `  x
)  =/=  Z  -> 
( f `  x
)  =  Z )
2019a1d 25 . . . . . . . . 9  |-  ( -.  ( f `  x
)  =/=  Z  -> 
( m  <  x  ->  ( f `  x
)  =  Z ) )
2120a1d 25 . . . . . . . 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 ) ) )
22 elfz2nn0 11478 . . . . . . . . 9  |-  ( x  e.  ( 0 ... m )  <->  ( x  e.  NN0  /\  m  e. 
NN0  /\  x  <_  m ) )
23 nn0re 10586 . . . . . . . . . . . . 13  |-  ( x  e.  NN0  ->  x  e.  RR )
24 nn0re 10586 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  ->  m  e.  RR )
25 lenlt 9451 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  m  e.  RR )  ->  ( x  <_  m  <->  -.  m  <  x ) )
2623, 24, 25syl2an 477 . . . . . . . . . . . 12  |-  ( ( x  e.  NN0  /\  m  e.  NN0 )  -> 
( x  <_  m  <->  -.  m  <  x ) )
27 pm2.21 108 . . . . . . . . . . . 12  |-  ( -.  m  <  x  -> 
( m  <  x  ->  ( f `  x
)  =  Z ) )
2826, 27syl6bi 228 . . . . . . . . . . 11  |-  ( ( x  e.  NN0  /\  m  e.  NN0 )  -> 
( x  <_  m  ->  ( m  <  x  ->  ( f `  x
)  =  Z ) ) )
29283impia 1184 . . . . . . . . . 10  |-  ( ( x  e.  NN0  /\  m  e.  NN0  /\  x  <_  m )  ->  (
m  <  x  ->  ( f `  x )  =  Z ) )
3029a1d 25 . . . . . . . . 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 ) ) )
3122, 30sylbi 195 . . . . . . . 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 ) ) )
3221, 31ja 161 . . . . . . 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 ) ) )
3332com12 31 . . . . . 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 ) ) )
3433ralimdva 2792 . . . . 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 ) ) )
3517, 34sylbid 215 . . . 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 ) ) )
3635ralimdva 2792 . . 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 ) ) )
3736reximdva 2826 . 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 ) ) )
381, 37syld 44 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3326   class class class wbr 4290    Fn wfn 5411   ` cfv 5416  (class class class)co 6089   supp csupp 6688    ^m cmap 7212   Fincfn 7308   finSupp cfsupp 7618   RRcr 9279   0cc0 9280    < clt 9416    <_ cle 9417   NN0cn0 10577   ...cfz 11435
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436
This theorem is referenced by:  pmatcoe1fsupp  30889
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