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Theorem fsum00 10856
Description: A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsumge0.1  |-  ( ph  ->  A  e.  Fin )
fsumge0.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
fsumge0.3  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
Assertion
Ref Expression
fsum00  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  = 
0 ) )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fsum00
Dummy variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsumge0.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  Fin )
21adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  A  e.  Fin )
3 fsumge0.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
43adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  B  e.  RR )
5 fsumge0.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
65adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  0  <_  B )
7 snssi 3581 . . . . . . . . . 10  |-  ( m  e.  A  ->  { m }  C_  A )
87adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  { m }  C_  A )
9 snfig 6531 . . . . . . . . . 10  |-  ( m  e.  A  ->  { m }  e.  Fin )
109adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  { m }  e.  Fin )
112, 4, 6, 8, 10fsumlessfi 10854 . . . . . . . 8  |-  ( (
ph  /\  m  e.  A )  ->  sum_ k  e.  { m } B  <_ 
sum_ k  e.  A  B )
1211adantlr 461 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  <_  sum_ k  e.  A  B
)
13 simpr 108 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  m  e.  A )
143, 5jca 300 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  ( B  e.  RR  /\  0  <_  B ) )
1514ralrimiva 2446 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B )
)
1615adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B ) )
17 nfcsb1v 2963 . . . . . . . . . . . . . 14  |-  F/_ k [_ m  /  k ]_ B
1817nfel1 2239 . . . . . . . . . . . . 13  |-  F/ k
[_ m  /  k ]_ B  e.  RR
19 nfcv 2228 . . . . . . . . . . . . . 14  |-  F/_ k
0
20 nfcv 2228 . . . . . . . . . . . . . 14  |-  F/_ k  <_
2119, 20, 17nfbr 3889 . . . . . . . . . . . . 13  |-  F/ k 0  <_  [_ m  / 
k ]_ B
2218, 21nfan 1502 . . . . . . . . . . . 12  |-  F/ k ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B )
23 csbeq1a 2941 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
2423eleq1d 2156 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( B  e.  RR  <->  [_ m  / 
k ]_ B  e.  RR ) )
2523breq2d 3857 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (
0  <_  B  <->  0  <_  [_ m  /  k ]_ B ) )
2624, 25anbi12d 457 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
( B  e.  RR  /\  0  <_  B )  <->  (
[_ m  /  k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2722, 26rspc 2716 . . . . . . . . . . 11  |-  ( m  e.  A  ->  ( A. k  e.  A  ( B  e.  RR  /\  0  <_  B )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2816, 27mpan9 275 . . . . . . . . . 10  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) )
2928simpld 110 . . . . . . . . 9  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  RR )
3029recnd 7516 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
31 sumsns 10809 . . . . . . . 8  |-  ( ( m  e.  A  /\  [_ m  /  k ]_ B  e.  CC )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
3213, 30, 31syl2anc 403 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
33 simplr 497 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  A  B  =  0 )
3412, 32, 333brtr3d 3874 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  <_  0 )
3528simprd 112 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  0  <_  [_ m  / 
k ]_ B )
36 0re 7488 . . . . . . 7  |-  0  e.  RR
37 letri3 7566 . . . . . . 7  |-  ( (
[_ m  /  k ]_ B  e.  RR  /\  0  e.  RR )  ->  ( [_ m  /  k ]_ B  =  0  <->  ( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  /  k ]_ B ) ) )
3829, 36, 37sylancl 404 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  =  0  <-> 
( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  / 
k ]_ B ) ) )
3934, 35, 38mpbir2and 890 . . . . 5  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  =  0
)
4039ralrimiva 2446 . . . 4  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
41 nfv 1466 . . . . 5  |-  F/ m  B  =  0
4217nfeq1 2238 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  =  0
4323eqeq1d 2096 . . . . 5  |-  ( k  =  m  ->  ( B  =  0  <->  [_ m  / 
k ]_ B  =  0 ) )
4441, 42, 43cbvral 2586 . . . 4  |-  ( A. k  e.  A  B  =  0  <->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
4540, 44sylibr 132 . . 3  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  B  = 
0 )
4645ex 113 . 2  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  ->  A. k  e.  A  B  =  0 ) )
47 isumz 10781 . . . . 5  |-  ( ( ( 0  e.  ZZ  /\  A  C_  ( ZZ>= ` 
0 )  /\  A. x  e.  ( ZZ>= ` 
0 )DECID  x  e.  A )  \/  A  e.  Fin )  ->  sum_ k  e.  A 
0  =  0 )
4847olcs 690 . . . 4  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
49 sumeq2 10748 . . . . 5  |-  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
5049eqeq1d 2096 . . . 4  |-  ( A. k  e.  A  B  =  0  ->  ( sum_ k  e.  A  B  =  0  <->  sum_ k  e.  A  0  =  0 ) )
5148, 50syl5ibrcom 155 . . 3  |-  ( A  e.  Fin  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
521, 51syl 14 . 2  |-  ( ph  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
5346, 52impbid 127 1  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  = 
0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   [_csb 2933    C_ wss 2999   {csn 3446   class class class wbr 3845   ` cfv 5015   Fincfn 6457   CCcc 7348   RRcr 7349   0cc0 7350    <_ cle 7523   ZZcz 8750   ZZ>=cuz 9019   sum_csu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-ico 9312  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by: (None)
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