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Theorem suppssfz 12144
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 7479 . . . . . . . 8  |-  ( F  e.  ( B  ^m  NN0 )  ->  F  Fn  NN0 )
42, 3syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  NN0 )
5 nn0ex 10842 . . . . . . . 8  |-  NN0  e.  _V
65a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
7 suppssfz.z . . . . . . 7  |-  ( ph  ->  Z  e.  V )
84, 6, 73jca 1177 . . . . . 6  |-  ( ph  ->  ( F  Fn  NN0  /\ 
NN0  e.  _V  /\  Z  e.  V ) )
98adantr 463 . . . . 5  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  ( F  Fn  NN0  /\  NN0  e.  _V  /\  Z  e.  V ) )
10 elsuppfn 6910 . . . . 5  |-  ( ( F  Fn  NN0  /\  NN0 
e.  _V  /\  Z  e.  V )  ->  (
n  e.  ( F supp 
Z )  <->  ( n  e.  NN0  /\  ( F `
 n )  =/= 
Z ) ) )
119, 10syl 17 . . . 4  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  (
n  e.  ( F supp 
Z )  <->  ( n  e.  NN0  /\  ( F `
 n )  =/= 
Z ) ) )
12 breq2 4399 . . . . . . . . . . . . 13  |-  ( x  =  n  ->  ( S  <  x  <->  S  <  n ) )
13 fveq2 5849 . . . . . . . . . . . . . 14  |-  ( x  =  n  ->  ( F `  x )  =  ( F `  n ) )
1413eqeq1d 2404 . . . . . . . . . . . . 13  |-  ( x  =  n  ->  (
( F `  x
)  =  Z  <->  ( F `  n )  =  Z ) )
1512, 14imbi12d 318 . . . . . . . . . . . 12  |-  ( x  =  n  ->  (
( S  <  x  ->  ( F `  x
)  =  Z )  <-> 
( S  <  n  ->  ( F `  n
)  =  Z ) ) )
1615rspcva 3158 . . . . . . . . . . 11  |-  ( ( n  e.  NN0  /\  A. x  e.  NN0  ( S  <  x  ->  ( F `  x )  =  Z ) )  -> 
( S  <  n  ->  ( F `  n
)  =  Z ) )
17 simplr 754 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  n  e.  NN0 )
18 suppssfz.s . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  NN0 )
1918adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  e.  NN0 )  ->  S  e.  NN0 )
2019adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  S  e.  NN0 )
21 nn0re 10845 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN0  ->  n  e.  RR )
22 nn0re 10845 . . . . . . . . . . . . . . . . . . . . 21  |-  ( S  e.  NN0  ->  S  e.  RR )
2318, 22syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  RR )
24 lenlt 9694 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  RR  /\  S  e.  RR )  ->  ( n  <_  S  <->  -.  S  <  n ) )
2521, 23, 24syl2anr 476 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( n  <_  S  <->  -.  S  <  n ) )
2625biimpar 483 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  n  <_  S )
27 elfz2nn0 11824 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 0 ... S )  <->  ( n  e.  NN0  /\  S  e. 
NN0  /\  n  <_  S ) )
2817, 20, 26, 27syl3anbrc 1181 . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( -.  S  <  n  ->  (
( F `  n
)  =/=  Z  ->  n  e.  ( 0 ... S ) ) ) )
31 eqneqall 2610 . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . 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 17 . . . . . . . . . 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 432 . . . . . . . . 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 48 . . . . . . . 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 427 . . . . . 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 427 . . . 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 3448 . 2  |-  ( (
ph  /\  A. x  e.  NN0  ( S  < 
x  ->  ( F `  x )  =  Z ) )  ->  ( F supp  Z )  C_  (
0 ... S ) )
461, 45mpdan 666 1  |-  ( ph  ->  ( F supp  Z ) 
C_  ( 0 ... S ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   _Vcvv 3059    C_ wss 3414   class class class wbr 4395    Fn wfn 5564   ` cfv 5569  (class class class)co 6278   supp csupp 6902    ^m cmap 7457   RRcr 9521   0cc0 9522    < clt 9658    <_ cle 9659   NN0cn0 10836   ...cfz 11726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727
This theorem is referenced by:  fsuppmapnn0fz  12146  fsfnn0gsumfsffz  17331
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