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Theorem fsuppmapnn0fiub0 12062
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 12061 . 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 3499 . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . 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 1017 . . . . . . . . . 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 10797 . . . . . . . . 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 1037 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  m  e.  NN0 )  /\  f  e.  M
)  ->  Z  e.  V )
13 suppvalfn 6905 . . . . . . . 8  |-  ( ( f  Fn  NN0  /\  NN0 
e.  _V  /\  Z  e.  V )  ->  (
f supp  Z )  =  {
x  e.  NN0  | 
( f `  x
)  =/=  Z }
)
149, 11, 12, 13syl3anc 1228 . . . . . . 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 3531 . . . . . 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 3577 . . . . . 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 2668 . . . . . . . . . . 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 11764 . . . . . . . . 9  |-  ( x  e.  ( 0 ... m )  <->  ( x  e.  NN0  /\  m  e. 
NN0  /\  x  <_  m ) )
23 nn0re 10800 . . . . . . . . . . . . 13  |-  ( x  e.  NN0  ->  x  e.  RR )
24 nn0re 10800 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  ->  m  e.  RR )
25 lenlt 9659 . . . . . . . . . . . . 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 1193 . . . . . . . . . 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 2872 . . . . 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 2872 . . 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 2938 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   supp csupp 6898    ^m cmap 7417   Fincfn 7513   finSupp cfsupp 7825   RRcr 9487   0cc0 9488    < clt 9624    <_ cle 9625   NN0cn0 10791   ...cfz 11668
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-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-int 4283  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-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  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:  pmatcoe1fsupp  18966
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