MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsuppmapnn0fiub0 Structured version   Unicode version

Theorem fsuppmapnn0fiub0 12102
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 12101 . 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 3494 . . . . . . . . . . . . . 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 7460 . . . . . . . . . . . . 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 10822 . . . . . . . . 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 6924 . . . . . . . 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 3526 . . . . . 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 3573 . . . . . 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 2658 . . . . . . . . . . 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 11795 . . . . . . . . 9  |-  ( x  e.  ( 0 ... m )  <->  ( x  e.  NN0  /\  m  e. 
NN0  /\  x  <_  m ) )
23 nn0re 10825 . . . . . . . . . . . . 13  |-  ( x  e.  NN0  ->  x  e.  RR )
24 nn0re 10825 . . . . . . . . . . . . 13  |-  ( m  e.  NN0  ->  m  e.  RR )
25 lenlt 9680 . . . . . . . . . . . . 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 2865 . . . . 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 2865 . . 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 2932 . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   class class class wbr 4456    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   supp csupp 6917    ^m cmap 7438   Fincfn 7535   finSupp cfsupp 7847   RRcr 9508   0cc0 9509    < clt 9645    <_ cle 9646   NN0cn0 10816   ...cfz 11697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698
This theorem is referenced by:  pmatcoe1fsupp  19329
  Copyright terms: Public domain W3C validator