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Theorem fsum00 12579
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 variable  m is distinct from all other variables.
StepHypRef Expression
1 fsumge0.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  Fin )
21adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  A  e.  Fin )
3 fsumge0.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
43adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  B  e.  RR )
5 fsumge0.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
65adantlr 697 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  0  <_  B )
7 snssi 3944 . . . . . . . . . 10  |-  ( m  e.  A  ->  { m }  C_  A )
87adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  { m }  C_  A )
92, 4, 6, 8fsumless 12577 . . . . . . . 8  |-  ( (
ph  /\  m  e.  A )  ->  sum_ k  e.  { m } B  <_ 
sum_ k  e.  A  B )
109adantlr 697 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  <_  sum_ k  e.  A  B
)
11 simpr 449 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  m  e.  A )
123, 5jca 520 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  ( B  e.  RR  /\  0  <_  B ) )
1312ralrimiva 2791 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B )
)
1413adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B ) )
15 nfcsb1v 3285 . . . . . . . . . . . . . 14  |-  F/_ k [_ m  /  k ]_ B
1615nfel1 2584 . . . . . . . . . . . . 13  |-  F/ k
[_ m  /  k ]_ B  e.  RR
17 nfcv 2574 . . . . . . . . . . . . . 14  |-  F/_ k
0
18 nfcv 2574 . . . . . . . . . . . . . 14  |-  F/_ k  <_
1917, 18, 15nfbr 4258 . . . . . . . . . . . . 13  |-  F/ k 0  <_  [_ m  / 
k ]_ B
2016, 19nfan 1847 . . . . . . . . . . . 12  |-  F/ k ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B )
21 csbeq1a 3261 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
2221eleq1d 2504 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( B  e.  RR  <->  [_ m  / 
k ]_ B  e.  RR ) )
2321breq2d 4226 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (
0  <_  B  <->  0  <_  [_ m  /  k ]_ B ) )
2422, 23anbi12d 693 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
( B  e.  RR  /\  0  <_  B )  <->  (
[_ m  /  k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2520, 24rspc 3048 . . . . . . . . . . 11  |-  ( m  e.  A  ->  ( A. k  e.  A  ( B  e.  RR  /\  0  <_  B )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2614, 25mpan9 457 . . . . . . . . . 10  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) )
2726simpld 447 . . . . . . . . 9  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  RR )
2827recnd 9116 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
29 sumsns 12538 . . . . . . . 8  |-  ( ( m  e.  A  /\  [_ m  /  k ]_ B  e.  CC )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
3011, 28, 29syl2anc 644 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
31 simplr 733 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  A  B  =  0 )
3210, 30, 313brtr3d 4243 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  <_  0 )
3326simprd 451 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  0  <_  [_ m  / 
k ]_ B )
34 0re 9093 . . . . . . 7  |-  0  e.  RR
35 letri3 9162 . . . . . . 7  |-  ( (
[_ m  /  k ]_ B  e.  RR  /\  0  e.  RR )  ->  ( [_ m  /  k ]_ B  =  0  <->  ( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  /  k ]_ B ) ) )
3627, 34, 35sylancl 645 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  =  0  <-> 
( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  / 
k ]_ B ) ) )
3732, 33, 36mpbir2and 890 . . . . 5  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  =  0
)
3837ralrimiva 2791 . . . 4  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
39 nfv 1630 . . . . 5  |-  F/ m  B  =  0
4015nfeq1 2583 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  =  0
4121eqeq1d 2446 . . . . 5  |-  ( k  =  m  ->  ( B  =  0  <->  [_ m  / 
k ]_ B  =  0 ) )
4239, 40, 41cbvral 2930 . . . 4  |-  ( A. k  e.  A  B  =  0  <->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
4338, 42sylibr 205 . . 3  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  B  = 
0 )
4443ex 425 . 2  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  ->  A. k  e.  A  B  =  0 ) )
45 sumz 12518 . . . . 5  |-  ( ( A  C_  ( ZZ>= ` 
0 )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
4645olcs 386 . . . 4  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
47 sumeq2 12490 . . . . 5  |-  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
4847eqeq1d 2446 . . . 4  |-  ( A. k  e.  A  B  =  0  ->  ( sum_ k  e.  A  B  =  0  <->  sum_ k  e.  A  0  =  0 ) )
4946, 48syl5ibrcom 215 . . 3  |-  ( A  e.  Fin  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
501, 49syl 16 . 2  |-  ( ph  ->  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  0 ) )
5144, 50impbid 185 1  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  <->  A. k  e.  A  B  = 
0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   [_csb 3253    C_ wss 3322   {csn 3816   class class class wbr 4214   ` cfv 5456   Fincfn 7111   CCcc 8990   RRcr 8991   0cc0 8992    <_ cle 9123   ZZ>=cuz 10490   sum_csu 12481
This theorem is referenced by:  ramcl  13399  jensen  20829  eqeelen  25845  axcgrid  25857  rrnmet  26540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ico 10924  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482
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