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Theorem suppssfz 30786
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 7235 . . . . . . . 8  |-  ( F  e.  ( B  ^m  NN0 )  ->  F  Fn  NN0 )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  NN0 )
5 nn0ex 10585 . . . . . . . 8  |-  NN0  e.  _V
65a1i 11 . . . . . . 7  |-  ( ph  ->  NN0  e.  _V )
7 suppssfz.z . . . . . . 7  |-  ( ph  ->  Z  e.  V )
84, 6, 73jca 1168 . . . . . 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 6698 . . . . 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 4296 . . . . . . . . . . . . 13  |-  ( x  =  n  ->  ( S  <  x  <->  S  <  n ) )
13 fveq2 5691 . . . . . . . . . . . . . 14  |-  ( x  =  n  ->  ( F `  x )  =  ( F `  n ) )
1413eqeq1d 2451 . . . . . . . . . . . . 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 3071 . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  e.  NN0 )  ->  S  e.  NN0 )
2019adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  n  e.  NN0 )  /\  -.  S  <  n )  ->  S  e.  NN0 )
21 nn0re 10588 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN0  ->  n  e.  RR )
22 nn0re 10588 . . . . . . . . . . . . . . . . . . . . 21  |-  ( S  e.  NN0  ->  S  e.  RR )
2318, 22syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  S  e.  RR )
24 lenlt 9453 . . . . . . . . . . . . . . . . . . . 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 11480 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 0 ... S )  <->  ( n  e.  NN0  /\  S  e. 
NN0  /\  n  <_  S ) )
2817, 20, 26, 27syl3anbrc 1172 . . . . . . . . . . . . . . . . 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 2705 . . . . . . . . . . . . . . . 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 3362 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   _Vcvv 2972    C_ wss 3328   class class class wbr 4292    Fn wfn 5413   ` cfv 5418  (class class class)co 6091   supp csupp 6690    ^m cmap 7214   RRcr 9281   0cc0 9282    < clt 9418    <_ cle 9419   NN0cn0 10579   ...cfz 11437
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438
This theorem is referenced by:  fsuppmapnn0fz  30797  fsfnn0gsumfsffz  30803
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