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Theorem fsum00 12532
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 452 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  A  e.  Fin )
3 fsumge0.2 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
43adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  B  e.  RR )
5 fsumge0.3 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  A )  ->  0  <_  B )
65adantlr 696 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  A )  /\  k  e.  A )  ->  0  <_  B )
7 snssi 3902 . . . . . . . . . 10  |-  ( m  e.  A  ->  { m }  C_  A )
87adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  A )  ->  { m }  C_  A )
92, 4, 6, 8fsumless 12530 . . . . . . . 8  |-  ( (
ph  /\  m  e.  A )  ->  sum_ k  e.  { m } B  <_ 
sum_ k  e.  A  B )
109adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  <_  sum_ k  e.  A  B
)
11 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  m  e.  A )
123, 5jca 519 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  A )  ->  ( B  e.  RR  /\  0  <_  B ) )
1312ralrimiva 2749 . . . . . . . . . . . 12  |-  ( ph  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B )
)
1413adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  ( B  e.  RR  /\  0  <_  B ) )
15 nfcsb1v 3243 . . . . . . . . . . . . . 14  |-  F/_ k [_ m  /  k ]_ B
1615nfel1 2550 . . . . . . . . . . . . 13  |-  F/ k
[_ m  /  k ]_ B  e.  RR
17 nfcv 2540 . . . . . . . . . . . . . 14  |-  F/_ k
0
18 nfcv 2540 . . . . . . . . . . . . . 14  |-  F/_ k  <_
1917, 18, 15nfbr 4216 . . . . . . . . . . . . 13  |-  F/ k 0  <_  [_ m  / 
k ]_ B
2016, 19nfan 1842 . . . . . . . . . . . 12  |-  F/ k ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B )
21 csbeq1a 3219 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  B  =  [_ m  /  k ]_ B )
2221eleq1d 2470 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  ( B  e.  RR  <->  [_ m  / 
k ]_ B  e.  RR ) )
2321breq2d 4184 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (
0  <_  B  <->  0  <_  [_ m  /  k ]_ B ) )
2422, 23anbi12d 692 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
( B  e.  RR  /\  0  <_  B )  <->  (
[_ m  /  k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2520, 24rspc 3006 . . . . . . . . . . 11  |-  ( m  e.  A  ->  ( A. k  e.  A  ( B  e.  RR  /\  0  <_  B )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) ) )
2614, 25mpan9 456 . . . . . . . . . 10  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  ( [_ m  / 
k ]_ B  e.  RR  /\  0  <_  [_ m  / 
k ]_ B ) )
2726simpld 446 . . . . . . . . 9  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  RR )
2827recnd 9070 . . . . . . . 8  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  e.  CC )
29 sumsns 12491 . . . . . . . 8  |-  ( ( m  e.  A  /\  [_ m  /  k ]_ B  e.  CC )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
3011, 28, 29syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  {
m } B  = 
[_ m  /  k ]_ B )
31 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  -> 
sum_ k  e.  A  B  =  0 )
3210, 30, 313brtr3d 4201 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  <_  0 )
3326simprd 450 . . . . . 6  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  0  <_  [_ m  / 
k ]_ B )
34 0re 9047 . . . . . . 7  |-  0  e.  RR
35 letri3 9116 . . . . . . 7  |-  ( (
[_ m  /  k ]_ B  e.  RR  /\  0  e.  RR )  ->  ( [_ m  /  k ]_ B  =  0  <->  ( [_ m  /  k ]_ B  <_  0  /\  0  <_  [_ m  /  k ]_ B ) ) )
3627, 34, 35sylancl 644 . . . . . 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 889 . . . . 5  |-  ( ( ( ph  /\  sum_ k  e.  A  B  =  0 )  /\  m  e.  A )  ->  [_ m  /  k ]_ B  =  0
)
3837ralrimiva 2749 . . . 4  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
39 nfv 1626 . . . . 5  |-  F/ m  B  =  0
4015nfeq1 2549 . . . . 5  |-  F/ k
[_ m  /  k ]_ B  =  0
4121eqeq1d 2412 . . . . 5  |-  ( k  =  m  ->  ( B  =  0  <->  [_ m  / 
k ]_ B  =  0 ) )
4239, 40, 41cbvral 2888 . . . 4  |-  ( A. k  e.  A  B  =  0  <->  A. m  e.  A  [_ m  / 
k ]_ B  =  0 )
4338, 42sylibr 204 . . 3  |-  ( (
ph  /\  sum_ k  e.  A  B  =  0 )  ->  A. k  e.  A  B  = 
0 )
4443ex 424 . 2  |-  ( ph  ->  ( sum_ k  e.  A  B  =  0  ->  A. k  e.  A  B  =  0 ) )
45 sumz 12471 . . . . 5  |-  ( ( A  C_  ( ZZ>= ` 
0 )  \/  A  e.  Fin )  ->  sum_ k  e.  A  0  = 
0 )
4645olcs 385 . . . 4  |-  ( A  e.  Fin  ->  sum_ k  e.  A  0  = 
0 )
47 sumeq2 12443 . . . . 5  |-  ( A. k  e.  A  B  =  0  ->  sum_ k  e.  A  B  =  sum_ k  e.  A  0 )
4847eqeq1d 2412 . . . 4  |-  ( A. k  e.  A  B  =  0  ->  ( sum_ k  e.  A  B  =  0  <->  sum_ k  e.  A  0  =  0 ) )
4946, 48syl5ibrcom 214 . . 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 184 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   [_csb 3211    C_ wss 3280   {csn 3774   class class class wbr 4172   ` cfv 5413   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946    <_ cle 9077   ZZ>=cuz 10444   sum_csu 12434
This theorem is referenced by:  ramcl  13352  jensen  20780  eqeelen  25747  axcgrid  25759  rrnmet  26428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435
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