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

Theorem suppssfz 12103
Description: Condition for a function over the nonnegative integers to have a support contained in a finite set of sequential integers. (Contributed by AV, 9-Oct-2019.)
Hypotheses
Ref Expression
suppssfz.z  |-  ( ph  ->  Z  e.  V )
suppssfz.f  |-  ( ph  ->  F  e.  ( B  ^m  NN0 ) )
suppssfz.s  |-  ( ph  ->  S  e.  NN0 )
suppssfz.b  |-  ( ph  ->  A. x  e.  NN0  ( S  <  x  -> 
( F `  x
)  =  Z ) )
Assertion
Ref Expression
suppssfz  |-  ( ph  ->  ( F supp  Z ) 
C_  ( 0 ... S ) )
Distinct variable groups:    x, F    x, S    x, Z
Allowed substitution hints:    ph( x)    B( x)    V( x)

Proof of Theorem suppssfz
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 suppssfz.b . 2  |-  ( ph  ->  A. x  e.  NN0  ( S  <  x  -> 
( F `  x
)  =  Z ) )
2 suppssfz.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( B  ^m  NN0 ) )
3 elmapfn 7460 . . . . . . . 8  |-  ( F  e.  ( B  ^m  NN0 )  ->  F  Fn  NN0 )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  NN0 )
5 nn0ex 10822 . . . . . . . 8  |-  NN0  e.  _V
65a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
7 suppssfz.z . . . . . . 7  |-  ( ph  ->  Z  e.  V )
84, 6, 73jca 1176 . . . . . 6  |-  ( ph  ->  ( F  Fn  NN0  /\ 
NN0  e.  _V  /\  Z  e.  V ) )
98adantr 465 . . . . 5  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  ( F  Fn  NN0  /\  NN0  e.  _V  /\  Z  e.  V ) )
10 elsuppfn 6925 . . . . 5  |-  ( ( F  Fn  NN0  /\  NN0 
e.  _V  /\  Z  e.  V )  ->  (
n  e.  ( F supp 
Z )  <->  ( n  e.  NN0  /\  ( F `
 n )  =/= 
Z ) ) )
119, 10syl 16 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  (
n  e.  ( F supp 
Z )  <->  ( n  e.  NN0  /\  ( F `
 n )  =/= 
Z ) ) )
12 breq2 4460 . . . . . . . . . . . . 13  |-  ( x  =  n  ->  ( S  <  x  <->  S  <  n ) )
13 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( x  =  n  ->  ( F `  x )  =  ( F `  n ) )
1413eqeq1d 2459 . . . . . . . . . . . . 13  |-  ( x  =  n  ->  (
( F `  x
)  =  Z  <->  ( F `  n )  =  Z ) )
1512, 14imbi12d 320 . . . . . . . . . . . 12  |-  ( x  =  n  ->  (
( S  <  x  ->  ( F `  x
)  =  Z )  <-> 
( S  <  n  ->  ( F `  n
)  =  Z ) ) )
1615rspcva 3208 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  /\  A. x  e.  NN0  ( S  <  x  ->  ( F `  x )  =  Z ) )  -> 
( S  <  n  ->  ( F `  n
)  =  Z ) )
17 simplr 755 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  n  e.  NN0 )
18 suppssfz.s . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  NN0 )
1918adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  e.  NN0 )  ->  S  e.  NN0 )
2019adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  S  e.  NN0 )
21 nn0re 10825 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN0  ->  n  e.  RR )
22 nn0re 10825 . . . . . . . . . . . . . . . . . . . . 21  |-  ( S  e.  NN0  ->  S  e.  RR )
2318, 22syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  RR )
24 lenlt 9680 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  RR  /\  S  e.  RR )  ->  ( n  <_  S  <->  -.  S  <  n ) )
2521, 23, 24syl2anr 478 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( n  <_  S  <->  -.  S  <  n ) )
2625biimpar 485 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  n  <_  S )
27 elfz2nn0 11795 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 0 ... S )  <->  ( n  e.  NN0  /\  S  e. 
NN0  /\  n  <_  S ) )
2817, 20, 26, 27syl3anbrc 1180 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  n  e.  ( 0 ... S ) )
2928a1d 25 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  -> 
( ( F `  n )  =/=  Z  ->  n  e.  ( 0 ... S ) ) )
3029ex 434 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( -.  S  <  n  ->  (
( F `  n
)  =/=  Z  ->  n  e.  ( 0 ... S ) ) ) )
31 eqneqall 2664 . . . . . . . . . . . . . . . 16  |-  ( ( F `  n )  =  Z  ->  (
( F `  n
)  =/=  Z  ->  n  e.  ( 0 ... S ) ) )
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( F `  n )  =  Z  ->  ( ( F `  n )  =/=  Z  ->  n  e.  ( 0 ... S
) ) ) )
3330, 32jad 162 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( S  <  n  ->  ( F `  n )  =  Z )  ->  (
( F `  n
)  =/=  Z  ->  n  e.  ( 0 ... S ) ) ) )
3433com23 78 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( F `  n )  =/=  Z  ->  ( ( S  <  n  ->  ( F `  n )  =  Z )  ->  n  e.  ( 0 ... S
) ) ) )
3534ex 434 . . . . . . . . . . . 12  |-  ( ph  ->  ( n  e.  NN0  ->  ( ( F `  n )  =/=  Z  ->  ( ( S  < 
n  ->  ( F `  n )  =  Z )  ->  n  e.  ( 0 ... S
) ) ) ) )
3635com14 88 . . . . . . . . . . 11  |-  ( ( S  <  n  -> 
( F `  n
)  =  Z )  ->  ( n  e. 
NN0  ->  ( ( F `
 n )  =/= 
Z  ->  ( ph  ->  n  e.  ( 0 ... S ) ) ) ) )
3716, 36syl 16 . . . . . . . . . 10  |-  ( ( n  e.  NN0  /\  A. x  e.  NN0  ( S  <  x  ->  ( F `  x )  =  Z ) )  -> 
( n  e.  NN0  ->  ( ( F `  n )  =/=  Z  ->  ( ph  ->  n  e.  ( 0 ... S
) ) ) ) )
3837ex 434 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( A. x  e.  NN0  ( S  <  x  ->  ( F `  x )  =  Z )  ->  (
n  e.  NN0  ->  ( ( F `  n
)  =/=  Z  -> 
( ph  ->  n  e.  ( 0 ... S
) ) ) ) ) )
3938pm2.43a 49 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( A. x  e.  NN0  ( S  <  x  ->  ( F `  x )  =  Z )  ->  (
( F `  n
)  =/=  Z  -> 
( ph  ->  n  e.  ( 0 ... S
) ) ) ) )
4039com23 78 . . . . . . 7  |-  ( n  e.  NN0  ->  ( ( F `  n )  =/=  Z  ->  ( A. x  e.  NN0  ( S  <  x  -> 
( F `  x
)  =  Z )  ->  ( ph  ->  n  e.  ( 0 ... S ) ) ) ) )
4140imp 429 . . . . . 6  |-  ( ( n  e.  NN0  /\  ( F `  n )  =/=  Z )  -> 
( A. x  e. 
NN0  ( S  < 
x  ->  ( F `  x )  =  Z )  ->  ( ph  ->  n  e.  ( 0 ... S ) ) ) )
4241com13 80 . . . . 5  |-  ( ph  ->  ( A. x  e. 
NN0  ( S  < 
x  ->  ( F `  x )  =  Z )  ->  ( (
n  e.  NN0  /\  ( F `  n )  =/=  Z )  ->  n  e.  ( 0 ... S ) ) ) )
4342imp 429 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  (
( n  e.  NN0  /\  ( F `  n
)  =/=  Z )  ->  n  e.  ( 0 ... S ) ) )
4411, 43sylbid 215 . . 3  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  (
n  e.  ( F supp 
Z )  ->  n  e.  ( 0 ... S
) ) )
4544ssrdv 3505 . 2  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  ( F supp  Z )  C_  (
0 ... S ) )
461, 45mpdan 668 1  |-  ( ph  ->  ( F supp  Z ) 
C_  ( 0 ... S ) )
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   _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   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-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-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-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  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:  fsuppmapnn0fz  12105  fsfnn0gsumfsffz  17138
  Copyright terms: Public domain W3C validator